Question

The solution of the differential equation $y-x\frac{dy}{dx}=a(y^2+\frac{dy}{dx})$ is

• A. $y=c(x+a)(1+ay)$
  
• B. $y=c(x+a)(1-ay)$
  
• C. $y=c(x-a)(1+ay)$
  
• D. $Noneofthese$

Answer 1

The correct option is B $y=c(x+a)(1-ay)$


$y-x\frac{dy}{dx}=a(y^2+\frac{dy}{dx})\Rightarrow y-a{y}^2=(x+a)\frac{dy}{dx}\Rightarrow \frac{dy}{y(1-ay)}=\frac{dx}{x+a}$
On integrating both sides, we get
$\Rightarrow logy-log(1-ay)=log(x+a)+logc$
$\Rightarrow \frac{y}{(1-ay)}=c(x+a)$ or c(x+a)(1-ay)=y