Question

Solution of $(x-y{)}^2\frac{dy}{dx}=a^2$ is:

• A. $2y=c+a\log \left(\frac{x-y-a}{x-y+a}\right)$
• B. $y=c+a\log \left(\frac{x-y+a}{x-y-a}\right)$
• C. $2y=c-a\log \left(\frac{x-y}{x+y}\right)$
• D. $2y^2=c+\log \left(\frac{x-y+a}{x-y-a}\right)$

Answer 1

The correct option is A $2y=c+a\log \left(\frac{x-y-a}{x-y+a}\right)$

${(x-y)}^2\frac{dy}{dx}=a^2\Rightarrow \frac{dy}{dx}=\frac{a^2}{{(x-y)}^2}$

Substituting $x=x-y\Rightarrow \frac{dv}{dx}=1-\frac{dy}{dx}$
$-\frac{dv}{dx}+1=\frac{a^2}{v^2}\Rightarrow \frac{dv}{dx}=\frac{-a^2+v^2}{v^2}\Rightarrow \frac{\frac{dv}{dx}{v}^2}{-a^2+v^2}=1$

Integarting both sides

$\int \frac{\frac{dv}{dx}{v}^2}{-a^2+v^2}dx=\int dx\Rightarrow \frac{a}{2}(-\log \left(\frac{x-y-a}{x-y+a}\right))+2y=c$

$\Rightarrow a\log \left(\frac{x-y-a}{x-y+a}\right)+c=2y$