Question

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

Answer 1

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

$\Rightarrow y-a{y}^2=(x+1)\frac{dy}{dx}$

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

$\Rightarrow In(x+1)=\int \frac{dy}{y(1-ay)}+C$

$\Rightarrow In(x+1)=\int \frac{dy}{y}+\int \frac{ady}{1-ay}+C$

$\Rightarrow In(x+1)Iny+a(-1)In(1-ay)$

$\Rightarrow (x+1)=In\frac{y}{(1-ay{)}^a}+C$

$\Rightarrow x+1=\frac{Ky}{(1-ay{)}^a}$