Question

Let the solution curve of the differential equation $x \frac{d y}{d x}-y=\sqrt{y^{2}+16 x^{2}}, y(1)=3$ be $y=y(x)$. Then $y (2)$ is equal to:

• A. 15
• B. 11
• C. 13
• D. 17

Answer 1

Answer: (A)

$y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
$\Rightarrow x \frac{d v}{d x}=\sqrt{v^{2}+16}$
$\Rightarrow \int \frac{d v}{\sqrt{v^{2}+16}}=\int \frac{d x}{x}$
$\Rightarrow \ln \left|v+\sqrt{v^{2}+16}\right|=\ln x+\ln C$
$\Rightarrow y+\sqrt{y^{2}+16 x^{2}}=C x^{2}$
As $y(1)=3 \Rightarrow C=8$
$\Rightarrow y(2)=15$