Class XII Chapter 9 – Differential Equations Maths

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Exercise 9.1

Question 1:

Determine order and degree(if defined) of differential equation Answer

The highest order derivative present in the differential equation is . Therefore, its order is four.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

Question 2:

Determine order and degree(if defined) of differential equation Answer

The given differential equation is:

The highest order derivative present in the differential equation is . Therefore, its order is one.

It is a polynomial equation in . The highest power raised to is 1. Hence, its degree is one.

Question 3:

Determine order and degree(if defined) of differential equation Answer

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The highest order derivative present in the given differential equation is . Therefore, its order is two.

It is a polynomial equation in and . The power raised to is 1. Hence, its degree is one.

Question 4:

Determine order and degree(if defined) of differential equation Answer

The highest order derivative present in the given differential equation is . Therefore, its order is 2.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

Question 5:

Determine order and degree(if defined) of differential equation Answer

The highest order derivative present in the differential equation is . Therefore, its order is two.

It is a polynomial equation in and the power raised to is 1. Hence, its degree is one.

Question 6:

Determine order and degree(if defined) of differential equation

Answer

The highest order derivative present in the differential equation is . Therefore, its order is three.

The given differential equation is a polynomial equation in . The highest power raised to is 2. Hence, its degree is 2.

Question 7:

Determine order and degree(if defined) of differential equation Answer

The highest order derivative present in the differential equation is . Therefore, its order is three.

It is a polynomial equation in . The highest power raised to is 1. Hence, its degree is 1.

Question 8:

Determine order and degree(if defined) of differential equation Answer

The highest order derivative present in the differential equation is . Therefore, its order is one.

The given differential equation is a polynomial equation in and the highest power raised to is one. Hence, its degree is one.

Question 9:

Determine order and degree(if defined) of differential equation Answer

The highest order derivative present in the differential equation is . Therefore, its order is two.

The given differential equation is a polynomial equation in and and the highest

power raised to is one. Hence, its degree is one.

Question 10:

Determine order and degree(if defined) of differential equation Answer

The highest order derivative present in the differential equation is . Therefore, its order is two.

This is a polynomial equation in and and the highest power raised to is one. Hence, its degree is one.

Question 11:

The degree of the differential equation

is

(A) 3 (B) 2 (C) 1 (D) not defined Answer

The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.

Hence, the correct answer is D.

Question 12:

The order of the differential equation

is

(A) 2 (B) 1 (C) 0 (D) not defined Answer

The highest order derivative present in the given differential equation is . Therefore, its order is two.

Hence, the correct answer is A.

Exercise 9.2

Question 1:

Answer

Differentiating both sides of this equation
with respect to *x*, we get:

Now, differentiating equation (1) with respect
to *x*, we get:

Substituting the values of in the given differential equation, we get the L.H.S. as:

Thus, the given function is the solution of the corresponding differential equation.

Question 2:

Answer

Differentiating both sides of this equation
with respect to *x*, we get:

Substituting the value of in the given differential equation, we get:

L.H.S. = = R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

Question 3:

Answer

Differentiating both sides of this equation
with respect to *x*, we get:

Substituting the value of in the given differential equation, we get:

L.H.S. = = R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

Question 4:

Answer

Differentiating both sides of the equation
with respect to *x*, we get:

L.H.S. = R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

Question 5:

Answer

Differentiating both sides with respect to *x*, we get:

Substituting the value of in the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

Question 6:

Answer

Differentiating both sides of this equation
with respect to *x*, we get:

Substituting the value of in the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

Question 7:

Answer

Differentiating both sides of this equation
with respect to *x*, we get:

L.H.S. = R.H.S.

Hence, the given function is the solution of the corresponding differential equation.

Question 8:

Answer

Differentiating both sides of the equation
with respect to *x*, we get:

Substituting the value of in equation (1), we get:

Hence, the given function is the solution of the corresponding differential equation.

Question 9:

Answer

Differentiating both sides of this equation
with respect to *x*, we get:

Substituting the value of in the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

Question 10:

Answer

Differentiating both sides of this equation
with respect to *x*, we get:

Substituting the value of in the given differential equation, we get:

Hence, the given function is the solution of the corresponding differential equation.

Question 11:

The numbers of arbitrary constants in the general solution of a differential equation of fourth order are:

(A) 0 (B) 2 (C) 3 (D) 4

Answer

We know that the number of constants in the general
solution of a differential equation
of order *n *is equal to its order.

Therefore, the number of constants in the general equation of fourth order differential equation is four.

Hence, the correct answer is D.

Question 12:

The numbers of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3 (B) 2 (C) 1 (D) 0

Answer

In a particular solution of a differential equation, there are no arbitrary constants. Hence, the correct answer is D.

Exercise 9.3

Question 1:

Answer

Differentiating both sides of the given
equation with respect
to *x*, we get:

Again, differentiating both sides
with respect to *x*, we get:

Hence, the required differential equation of the given curve is

Question 2:

Answer

Differentiating both sides with respect to *x*, we get:

Again, differentiating both sides
with respect to *x*, we get:

Dividing equation (2) by equation (1), we get:

This is the required differential equation of the given curve.

Question 3:

Answer

Differentiating both sides with respect to *x*, we get:

Again, differentiating both sides
with respect to *x*, we get:

Multiplying equation (1) with (2) and then adding it to equation (2), we get:

Now, multiplying equation (1) with equation (3) and subtracting equation (2) from it, we get:

Substituting the values of in equation (3), we get:

This is the required differential equation of the given curve.

Question 4:

Answer

Differentiating both sides with respect to *x*, we get:

Multiplying equation (1) with equation (2) and then subtracting it from equation (2), we get:

Differentiating both sides with respect to *x*, we get:

Dividing equation (4) by equation (3), we get:

This is the required differential equation of the given curve.

Question 5:

Answer

Differentiating both sides with respect to *x*, we get:

Again, differentiating with respect
to *x*, we get:

Adding equations (1) and (3), we get:

This is the required differential equation of the given curve.

Question 6:

Form the differential equation
of the family of circles
touching the *y*-axis at the origin.
Answer

The centre of the circle
touching the *y*-axis at origin lies on the *x*-axis. Let (*a*, 0) be the centre of the
circle.

Since it touches the *y*-axis at origin, its radius is *a*.

Now, the equation of the circle
with centre (*a*, 0) and radius
(*a) *is

Differentiating equation
(1) with respect
to *x*, we get:

Now, on substituting the value of *a *in equation (1), we get:

This is the required differential equation.

Question 7:

Form the differential equation
of the family of parabolas having vertex at origin and axis along
positive *y*-axis.

Answer

The equation of the parabola having the vertex at origin and the axis along the positive

*y*-axis is:

Differentiating equation
(1) with respect
to *x*, we get:

Dividing equation (2) by equation (1), we get:

This is the required differential equation.

Question 8:

Form the differential equation of the family
of ellipses having
foci on *y*-axis and centre at origin.

Answer

The equation of the family of ellipses having foci on the

Differentiating equation
(1) with respect
to *x*, we get:

Again, differentiating with respect
to *x*, we get:

Substituting this value in equation (2), we get:

This is the required differential equation.

Question 9:

Form the differential equation of the family of hyperbolas having
foci on *x*-axis and centre
at origin.

Answer

The equation of the family of hyperbolas with the centre at origin and foci along the

Differentiating both sides of equation (1) with respect
to *x*, we get:

Again, differentiating both sides
with respect to *x*, we get:

Substituting the value of in equation (2), we get:

This is the required differential equation.

Question 10:

Form the differential equation of the family of circles having
centre on *y*-axis and radius 3 units.

Answer

Let the centre of the circle on *y*-axis be (0, *b*).

The differential equation of the family of circles with centre at (0,

Differentiating equation
(1) with respect
to *x*, we get:

Substituting the value of (*y *– *b*) in equation (1), we get:

This is the required differential equation.

Question 11:

Which of the following differential equations has as the general solution?

A.

B.

C.

D.

Answer

The given equation is:

Differentiating with respect to *x*, we get:

Again, differentiating with respect
to *x*, we get:

This is the required differential equation of the given equation of curve. Hence, the correct answer is B.

Question 12:

Which of the following differential equation has as one of its particular solution?

A.

B.

C.

D.

Answer

The given equation of curve is

Again, differentiating with respect
to *x*, we get:

Now, on substituting the values of *y*, from equation
(1) and (2) in each of

the given alternatives, we find that only the differential equation given in alternative C is correct.

Hence, the correct answer is C.

Exercise 9.4

Question 1:

Answer

The given differential equation is:

Now, integrating both sides of this equation, we get:

This is the required general solution of the given differential equation.

Question 2:

Answer

The given differential equation is:

Now, integrating both sides of this equation, we get:

This is the required general solution of the given differential equation.

Question 3:

Answer

The given differential equation is:

Now, integrating both sides, we get:

This is the required general solution of the given differential equation.

Question 4:

Answer

The given differential equation is:

Integrating both sides of this equation, we get:

Substituting these values in equation (1), we get:

This is the required general solution of the given differential equation.

Question 5:

Answer

The given differential equation is:

Integrating both sides of this equation, we get:

Let (*e ^{x} *+

Differentiating both sides with respect to *x*, we get:

Substituting this value in equation (1), we get:

This is the required general solution of the given differential equation.

Question 6:

Answer

The given differential equation is:

Integrating both sides of this equation, we get:

This is the required general solution of the given differential equation.

Question 7:

Answer

The given differential equation is:

Integrating both sides, we get:

Substituting this value in equation (1), we get:

This is the required general solution of the given differential equation.

Question 8:

Answer

The given differential equation is:

Integrating both sides, we get:

This is the required general solution of the given differential equation.

Question 9:

Answer

The given differential equation is:

Integrating both sides, we get:

Substituting this value in equation (1), we get:

This is the required general solution of the given differential equation.

Question 10:

Answer

The given differential equation is:

Integrating both sides, we get:

Substituting the values of in equation (1), we get:

This is the required general solution of the given differential equation.

Question 11:

Answer

The given differential equation is:

Integrating both sides, we get:

Comparing the coefficients of *x*^{2} and *x*, we get:

*A *+ *B *= 2 *B *+ *C *= 1 *A *+ *C *= 0

Solving these equations, we get:

Substituting the values of A, B, and C in equation (2), we get:

Therefore, equation (1) becomes:

Substituting C = 1 in equation (3), we get:

Question 12:

Answer

Integrating both sides, we get:

Comparing the coefficients of *x*^{2}, *x,
*and constant, we get:

Solving these equations, we get
Substituting the values of *A*, *B, *and *C *in equation
(2), we get:

Therefore, equation (1) becomes:

Substituting the value of *k*^{2} in equation (3), we get:

Question 13:

Answer

Integrating both sides, we get:

Substituting C = 1 in equation (1), we get:

Question 14:

Answer

Integrating both sides, we get:

Substituting C = 1 in equation (1), we get:

*y *= sec *x*

* *

* *

Question 15:

Find the equation of a curve passing through the point (0, 0) and whose differential

equation is .

Answer

The differential equation of the curve is:

Integrating both sides, we get:

Substituting this value in equation (1), we get:

Now, the curve passes through point (0, 0).

Substituting in equation (2), we get:

Hence, the required equation of the curve is

Question 16:

For the differential equation find the solution curve passing through the point (1, –1).

Answer

The differential equation of the given curve is:

Integrating both sides, we get:

Now, the curve passes through point (1, –1).

Substituting C = –2 in equation (1), we get:

This is the required solution of the given curve.

Question 17:

Find the equation of a curve passing through the point (0, –2) given that at any point

on the curve,
the product of the slope of its tangent and *y*-coordinate of the point
is equal to the *x*-coordinate of the
point.

Answer

Let *x *and *y *be the *x*-coordinate and *y*-coordinate of the curve respectively.

We know that the slope of a tangent to the curve in the coordinate axis is given by the

According to the given information, we get:

Integrating both sides, we get:

Now, the curve passes through point (0, –2).

∴ (–2)^{2} – 0^{2} = 2C

⇒ 2C = 4

Substituting 2C = 4 in equation (1), we get:

*y*^{2} – *x*^{2}
= 4

This is the required equation of the curve.

Question 18:

At any point (*x*,
*y*) of a curve, the slope of the tangent is twice the slope of the line segment
joining the point of contact
to the point (–4, –3). Find the equation of the curve
given that it passes through (–2, 1).

Answer

It is given that (*x*, *y*)
is the point of contact of the curve and its tangent.

The slope (*m*_{1}) of the line segment joining
(*x*, *y*) and (–4, –3) is

We know that the slope of the tangent to the curve is given by the relation,

According to the given information:

Integrating both sides, we get:

This is the general equation of the curve.

It is given that it passes through point (–2, 1).

Substituting C = 1 in equation (1), we get:

*y *+ 3 = (*x *+ 4)^{2}

This is the required equation of the curve.

Question 19:

The volume
of spherical balloon
being inflated changes
at a constant rate. If initially its radius
is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after *t *seconds.

Answer

Let the rate of change of the volume
of the balloon be *k *(where *k *is a constant).

Integrating both sides, we get:

⇒ 4π × 3^{3} = 3 (*k *× 0 + C)

⇒ 108π = 3C

⇒ C = 36π

At *t *= 3, *r *= 6:

⇒ 4π × 6^{3} = 3
(*k *× 3 + C)

⇒ 864π = 3 (3*k *+ 36π)

⇒ 3*k *= –288π – 36π = 252π

⇒ *k *= 84π

Substituting the values of *k *and
C in equation (1), we get:

Thus, the radius of the balloon
after *t *seconds is .

Question 20:

In a bank, principal increases continuously at the rate of *r*%
per year. Find the value of *r*

if Rs 100 doubles itself in 10 years (log* _{e} *2 = 0.6931).

Answer

Let

Integrating both sides, we get:

It is given that when *t *= 0, *p *=
100.

⇒ 100 = *e ^{k} *… (2)

Now, if

Hence, the value of *r *is 6.93%.

Question 21:

In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs

1000 is deposited with this bank, how much will it worth after 10 years . Answer

Let *p *and *t *be the principal and time respectively.

It is given that the principal increases continuously at the rate of 5% per year.

Integrating both sides, we get:

Now, when *t *= 0, *p *= 1000.

⇒ 1000 = *e*^{C} … (2)

At *t *= 10, equation (1) becomes:

Hence, after 10 years the amount will worth Rs 1648.

Question 22:

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

Answer

Let *y *be the number of bacteria at any instant
*t*.

It is given that the rate of growth of the bacteria is proportional to the number present.

Integrating both sides, we get:

Let *y*_{0} be the number of bacteria at *t *= 0.

⇒ log *y*_{0} = C

Substituting the value of C in equation (1), we get:

Also, it is given that the number of bacteria increases by 10% in 2 hours.

Substituting this value in equation (2), we get:

Therefore, equation (2) becomes:

Now, let the time when the number of bacteria increases from 100000 to 200000 be *t*_{1}.

⇒ *y *= 2*y*_{0} at *t
*= *t*_{1}

From equation (4), we get:

Hence, in hours the number of bacteria increases from 100000 to 200000.

Question 23:

The general solution of the differential equation

A.

B.

C.

D.

Answer

Integrating both sides, we get:

Hence, the correct answer is A.

Exercise 9.5

Question 1:

Answer

The given differential equation
i.e., (*x*^{2} + *xy*) *dy *= (*x*^{2} + *y*^{2}) *dx *can be written as:

This shows that equation (1) is a homogeneous equation. To solve it, we make the substitution as:

*y *= *vx*

Differentiating both sides with respect to *x*, we get:

Substituting the values of *v *and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 2:

Answer

The given differential equation is:

Thus, the given equation is a homogeneous equation. To solve it, we make the substitution as:

*y *=
*vx*

Differentiating both sides with respect to *x*, we get:

Substituting the values of

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 3:

Answer

The given differential equation is:

Thus, the given differential equation is a homogeneous equation. To solve it, we make the substitution as:

*y *= *vx*

* *

* *

Substituting the values of *y *and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 4:

Answer

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation. To solve it, we make the substitution as:

*y *= *vx*

* *

* *

* *

Substituting the values of *y *and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 5:

Answer

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation. To solve it, we make the substitution as:

*y *= *vx*

* *

Substituting the values of

Integrating both sides, we get:

This is the required solution for the given differential equation.

Question 6:

Answer

Therefore, the given differential equation is a homogeneous equation. To solve it, we make the substitution as:

*y *=
*vx*

* *

* *

Substituting the values of *v *and
in equation
(1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 7:

Answer

The given differential equation is:

*y *= *vx*

* *

* *

Substituting the values of *y *and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 8:

Answer

*y *= *vx*

* *

* *

Substituting the values of *y *and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 9:

Answer

*y *= *vx*

* *

* *

* *

Substituting the values
of *y *and
in equation
(1), we get:

Integrating both sides, we get:

Therefore, equation (1) becomes:

This is the required solution of the given differential equation.

Question 10:

Answer

*x *= *vy*

* *

Substituting the values of *x *and
in equation
(1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Question 11:

Answer

*y *= *vx*

* *

* *

Substituting the values of *y *and in equation (1), we get:

Integrating both sides, we get:

Now, *y *= 1 at *x *= 1.

Substituting the value of 2*k *in
equation (2), we get:

This is the required solution of the given differential equation.

Question 12:

Answer

*y *= *vx*

* *

* *

Substituting the values of *y *and in equation (1), we get:

Integrating both sides, we get:

Now, *y *= 1 at *x *= 1.

in equation (2), we get:

This is the required solution of the given differential equation.

Question 13:

Answer

Therefore, the given differential equation is a homogeneous equation. To solve this differential equation, we make the substitution as:

*y *= *vx*

* *

Substituting the values of

Integrating both sides, we get:

Now,

.

Substituting C = *e *in equation
(2), we get:

This is the required solution of the given differential equation.

Question 14:

Answer

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *=
*vx*

* *

Substituting the values of

Integrating both sides, we get:

This is the required
solution of the given differential equation. Now, *y *=
0 at *x *= 1.

Substituting C = *e *in equation
(2), we get:

This is the required solution of the given differential equation.

Question 15:

Answer

*y *= *vx*

* *

Substituting the value of

Integrating both sides, we get:

Now, *y *= 2 at *x *= 1.

Substituting *C *= –1 in equation
(2), we get:

This is the required solution of the given differential equation.

Question 16:

A homogeneous differential equation of the form can be solved by making the substitution

*A. **y *= *vx*

*B. **v *= *yx*

*C. **x *= *vy*

*D. **x *= *v*

Answer

For solving
the homogeneous equation
of the form , we need to make the substitution as *x *= *vy*.

Hence, the correct answer is C.

Question 17:

Which of the following is a homogeneous differential equation?

A.

B.

C.

D.

Answer

Function F(*x*, *y*) is said to be the homogenous function of degree
*n, *if F(λ*x*, λ*y*) =
λ* ^{n} *F(

Consider the equation given in alternativeD:

Hence, the differential equation given in alternative D is a homogenous equation.

Exercise 9.6

Question 1:

Answer

The given differential equation is

This is in the form of

The solution of the given differential equation is given by the relation,

Therefore, equation (1) becomes:

This is the required general solution of the given differential equation.

Question 2:

Answer

The given differential equation is

The solution of the given differential equation is given by the relation,

This is the required general solution of the given differential equation.

Question 3:

Answer

The given differential equation is:

The solution of the given differential equation is given by the relation,

This is the required general solution of the given differential equation.

Question 4:

Answer

The given differential equation is:

The general solution of the given differential equation is given by the relation,

Question 5:

Answer

By second fundamental theorem of calculus, we obtain

Question 6:

Answer

The given differential equation is:

This equation is in the form of a linear differential equation as:

The general solution of the given differential equation is given by the relation,

Question 7:

Answer

The given differential equation is:

This equation is the form of a linear differential equation as:

The general solution of the given differential equation is given by the relation,

Substituting the value of in equation (1), we get:

This is the required general solution of the given differential equation.

Question 8:

Answer

This equation is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Question 9:

Answer

This equation is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Question 10:

Answer

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Question 11:

Answer

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Question 12:

Answer

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Question 13:

Answer

The given differential equation is This is a linear equation of the form:

The general solution of the given differential equation is given by the relation,

Now, Therefore,

Substituting C = –2 in equation (1), we get:

Hence, the required solution of the given differential equation is

Question 14:

Answer

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Now, *y *= 0 at *x *= 1. Therefore,

Substituting in equation (1), we get:

This is the required general solution of the given differential equation.

Question 15:

Answer

The given differential equation is This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Now, Therefore, we get:

Substituting C = 4 in equation (1), we get:

This is the required particular solution of the given differential equation.

Question 16:

Find the equation
of a curve passing through the origin given that the slope of the tangent
to the curve at any point (*x*, *y*) is equal to the sum
of the coordinates of the point.

Answer

Let *F *(*x*, *y*)
be the curve passing through
the origin.

At point
(*x*, *y*), the slope of the curve will be According to the given information:

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Substituting in equation (1), we get:

The curve passes through the origin. Therefore, equation (2) becomes:

1 = C

Substituting C = 1 in equation (2), we get:

⇒

Hence, the required equation of curve passing through the origin is

Question 17:

Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

Answer

Let *F *(*x*, *y*)
be the curve and let (*x*, *y*) be a point on the curve. The slope of the tangent

to the curve at (

This is a linear differential equation of the form:

The general equation of the curve is given by the relation,

Therefore, equation (1) becomes:

The curve passes through point (0, 2). Therefore, equation (2) becomes:

0 + 2 – 4 = C*e*^{0}

⇒ – 2 = C

⇒ C = – 2

Substituting C = –2 in equation (2), we get:

This is the required equation of the curve.

Question 18:

The integrating factor of the differential equation is

*A. **e*^{–x}

*B. **e*^{–y}

C.

D. *x*

Answer

The given differential equation is:

This is a linear differential equation of the form:

The integrating factor (I.F) is given by the relation,

Hence, the correct answer is C.

Question 19:

The integrating factor of the differential equation.

is

A.

B.

C.

D.

Answer

The given differential equation is:

This is a linear differential equation of the form:

The integrating factor (I.F) is given by the relation,

Hence, the correct answer is D.

Miscellaneous Solutions

Question 1:

For each of the differential equations given below, indicate its order and degree (if defined).

(i)

(ii)

(iii)

Answer

(i) The differential equation is given as:

The highest order derivative present in the differential equation is . Thus, its order is two. The highest power raised to is one. Hence, its degree is one.

(ii)

The differential equation is given as:

The highest order derivative present in the differential equation is . Thus, its order is

one. The highest power raised to is three. Hence, its degree is three.

(iii) The differential equation is given as:

The highest order derivative present in the differential equation is . Thus, its order is four.

However, the given differential equation is not a polynomial equation. Hence, its degree is not defined.

Question 2:

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i)

(ii)

(iii)

(iv)

Answer

(i)

Differentiating both sides with respect to *x*, we get:

Again, differentiating both sides
with respect to *x*, we get:

Now, on substituting the values of and in the differential equation, we get:

⇒ L.H.S. ≠ R.H.S.

Hence, the given function is not a solution of the corresponding differential equation.

(ii)

Differentiating both sides with respect to *x*, we get:

Again, differentiating both sides
with respect to *x*, we get:

Now, on substituting the values of and in the L.H.S. of the given differential equation, we get:

Hence, the given function is a solution of the corresponding differential equation.

(iii)

Differentiating both sides with respect to *x*, we get:

Again, differentiating both sides
with respect to *x*, we get:

Substituting the value of in the L.H.S. of the given differential equation, we get:

Hence, the given function is a solution of the corresponding differential equation.

(iv)

Differentiating both sides with respect to *x*, we get:

Substituting the value of in the L.H.S. of the given differential equation, we get:

Hence, the given function is a solution of the corresponding differential equation.

Question 3:

Form the differential equation representing the family of curves given by

where

Differentiating with respect to *x*, we get:

From equation (1), we get:

On substituting this value in equation (3), we get:

Hence, the differential equation of the family of curves is given as

Question 4:

Prove that is the general solution of differential

equation , where

This is a homogeneous equation. To simplify it, we need to make the substitution as:

Substituting the values of *y *and in equation (1), we get:

Integrating both sides, we get:

Substituting the values of *I*_{1} and *I*_{2} in equation (3), we get:

Therefore, equation (2) becomes:

Hence, the given result is proved.

Question 5:

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Answer

The equation of a circle in the first quadrant with centre (

Differentiating equation
(1) with respect
to *x*, we get:

Substituting the value of *a *in
equation (1), we get:

Hence, the required differential equation of the family of circles is

Question 6:

Find the general solution of the differential equation Answer

Integrating both sides, we get:

Question 7:

Show that the general
solution of the differential equation
is given by (*x
*+ *y *+ 1) = *A *(1
– *x *– *y *– 2*xy*), where *A
*is parameter

Answer

Integrating both sides, we get:

Hence, the given result is proved.

Find the equation of the curve passing through the point whose differential equation is,

Answer

The differential equation of the given curve is:

Integrating both sides, we get:

The curve passes through point

On substituting in equation (1), we get:

Hence, the required equation of the curve is

Question 9:

Find the particular solution of the differential equation

, given that

Integrating both sides, we get:

Substituting these values in equation (1), we get:

Now, *y *= 1 at *x *= 0.

Therefore, equation (2) becomes:

Substituting in equation (2), we get:

This is the required particular solution of the given differential equation.

Question 10:

Solve the differential equation

Answer

Differentiating it with respect
to *y*, we get:

From equation (1) and equation (2), we get:

Integrating both sides, we get:

Question 11:

Find a particular solution of the differential equation , given that

Substituting the values of *x *– *y *and in equation (1), we get:

Integrating both sides, we get:

Now, *y *= –1 at *x *= 0.

Therefore, equation (3) becomes:

log 1 = 0 – 1 + C

⇒ C = 1

Substituting C = 1 in equation (3) we get:

This is the required particular solution of the given differential equation.

Question 12:

Solve the differential equation Answer

This equation is a linear differential equation of the form

The general solution of the given differential equation is given by,

Question 13:

Find a particular solution of the differential equation ,

given that *y *= 0 when Answer

The given differential equation is:

This equation is a linear differential equation of the form

The general solution of the given differential equation is given by,

Now,

Therefore, equation (1) becomes:

in equation (1), we get:

This is the required particular solution of the given differential equation.

Question 14:

Find a particular solution
of the differential equation , given
that *y *=
0 when *x *= 0

Answer

Integrating both sides, we get:

Substituting this value in equation (1), we get:

Now, at *x *=
0 and *y *= 0, equation (2) becomes:

Substituting C = 1 in equation (2), we get:

This is the required particular solution of the given differential equation.

Question 15:

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Answer

Let the population at any instant
(*t) *be *y*.

It is given that the rate of increase of population is proportional to the number of inhabitants at any instant.

Integrating both sides, we get: log *y *= *kt *+ C … (1)

In the year 1999,
*t *= 0 and *y *= 20000.
Therefore, we get:

log 20000 = C … (2)

In the year 2004, *t *= 5 and *y *= 25000. Therefore, we get:

In the year 2009, *t *= 10 years.

Now, on substituting the values of *t*, *k, *and C in equation
(1), we get:

Hence, the population of the village in 2009 will be 31250.

Question 16:

The general solution of the differential equation is

A. *xy *= C

B. *x *= C*y*^{2}

*C. **y *= C*x*

D. *y *= C*x*^{2} Answer

The given differential equation is:

Integrating both sides, we get:

Hence, the correct answer is C.

Question 17:

The general solution of a differential equation of the type is

A.

B.

C.

D.

Answer

The integrating factor of the given differential equation The general solution of the differential equation is given by,

Hence, the correct answer is C.

Question 18:

The general solution of the differential equation is

A. *xe ^{y} *+

B. *xe ^{y} *+

C. *ye ^{x} *+

D. *ye ^{y} *+

The given differential equation is:

This is a linear differential equation of the form

The general solution of the given differential equation is given by,

Hence, the correct answer is C.