Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e2,k)$. If the curve satisfies the differential equation $2(3+y)e^{2x}dx-(7+e^{2x})dy=0$, then $k$ is equal to

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $\frac{2e^{2x}}{7+e^{2x}}dx=\frac{dy}{3+y}\Rightarrow\ln(7+e^{2x})=\ln(3+y)+C$. At $(0,5)$: $\ln8=\ln8+C$, so $C=0$ and $3+y=7+e^{2x}$. At $x=\ln2$: $3+k=7+4$, so $k=8$. Answer (3).

Q. Let a curve $y=f(x)$ pass through the points $(0,5)$ and $(\log_e2,k)$. If the curve satisfies the differential equation $2(3+y)e^{2x}dx-(7+e^{2x})dy=0$, then $k$ is equal to

Explanation

Reviewed by Dr. Abubacker

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