If $x\ \ dy = y(dx + y\ dy),\ y(1) = 1$ and $y(x) 0$, then $y( - 3)$ is equal to

Question

If $x\ \ dy = y(dx + y\ dy),\ y(1) = 1$ and $y(x) > 0$, then $y( - 3)$ is equal to

Dr. Abubacker · Accepted Answer

(a)
Given, $\frac{x\ \ dy - y\ \ dx}{y^{2}} = dy$
$\Rightarrow$ $d\left( \frac{x}{y} ight) = - dy$
$\Rightarrow$ $\frac{x}{y} = - y + c$ [integrating]
As $y(1) = 1\ \  \Rightarrow c = 2$
$	herefore$ $\frac{x}{y} + y = 2$
Again, for $x = - 3$
$- 3 + y^{2} = 2y$
$\Rightarrow$ $(y + 1)(y - 3) = 0$
Also, $y > 0$
$\Rightarrow$ $y = 3$ [neglecting $y = - 1$]

Q. If $x\ \ dy = y(dx + y\ dy),\ y(1) = 1$ and $y(x) > 0$, then $y( - 3)$ is equal to

Explanation

Reviewed by Dr. Abubacker

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