Question

Find the general solution of given differential equation.

$x^2\frac{dy}{dx}=\frac{y(x+y)}{2}$

• A. $(y-x{)}^2=c{y}^{2}x$
• B. $(y+x{)}^2=c{y}^{2}x$
• C. $(y+x{)}^2=c{x}^{2}x$
• D. $(x-y{)}^2=c{y}^{2}x$

Answer 1

Answer: (A), (D)

$(y-x{)}^2=c{y}^{2}x$
$(x-y{)}^2=c{y}^{2}x$

$x^2\frac{dy}{dx}=\frac{y(x+y)}{2}\Rightarrow \frac{dy}{dx}=\frac{\left(y/x\right)(1+y/x)}{2}$
Substitute $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$
$\Rightarrow v+x\frac{dv}{dx}=\frac{v(1+v)}{2}\Rightarrow x\frac{dv}{dx}=\frac{v^2-v}{2}$
$\Rightarrow \frac{dv}{v(v-1)}=\frac{dx}{2x}\Rightarrow \frac{dv}{v-1}-\frac{dv}{v}=\frac{dx}{2x}$
Integrating we get, $\Rightarrow \log \frac{v-1}{v}=\frac{1}{2}\log x+\log k$
$\Rightarrow 1-\frac{1}{v}=k\sqrt{x}\Rightarrow (x-y{)}^2=c{y}^{2}x$