Question

The solution of the differential equation $\frac{x\frac{dy}{dx}-y}{\sqrt{x^{2}-y^{2}}}=10x^{2}is$

• A. $sin^{-1}\left(\frac{y}{x}\right)-5x^{2} =C$
• B. $sin^{-1}\left(\frac{y}{x}\right)=10x^{2} +C$
• C. $\frac{y}{x}=5x^{2}+C$
• D. $sin^{-1}\left(\frac{y}{x}\right)=10x^{2} +Cx$
• E. $sin^{-1}\left(\frac{y}{x}\right)+5x^{2} =C$

Answer 1

Answer: (A)

Given, $\frac{x \frac{d y}{d x}-y}{\sqrt{x^{2}-y^{2}}}=10 x^{2}$
$\Rightarrow \frac{1}{x^{2}} \frac{1}{\sqrt{x^{2}-y^{2}}}\left(x \frac{d y}{d x}-y\right)=10$
$\Rightarrow \frac{1}{x^{2}} \cdot \frac{1}{x \sqrt{1-\left(\frac{y}{x}\right)^{2}}}\left(\frac{x d y-y d x}{d x}\right)=10$
$\Rightarrow \frac{1}{\sqrt{1-\left(\frac{y}{x}\right)^{2}}}\left(\frac{x d y-y d x}{x^{2}}\right)=10 x d x$
$\Rightarrow \int d\left(\sin ^{-1} \frac{y}{x}\right)=\int 10 x d x$
$\Rightarrow \sin ^{-1} \frac{y}{x}=\frac{10 x^{2}}{2}+C$
$\Rightarrow \sin ^{-1} \frac{y}{x}=5 x^{2}+C$
$\therefore \sin ^{-1} \frac{y}{x}-5 x^{2}=C$