Question

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

• A. $y=c(1-ay)(x+a)$
• B. $2y=c(1-ay)(x+a)$
• C. $y=c(1-ay)(x+a)/2$
• D. $2y=c(1+ay)(x+a)$

Answer 1

Answer: (A), (B), (C)

$y=c(1-ay)(x+a)$
$y=c(1-ay)(x+a)/2$
$2y=c(1-ay)(x+a)$

Given, $y-\frac{xdy}{dx}=a{y}^2+\frac{a.dy}{dx}$
$y-a{y}^2=(x+a).\frac{dy}{dx}$
$\frac{dx}{x+a}=\frac{dy}{y(1-ay)}$
$\int \frac{dx}{x+a}=\int \frac{1-ay+ay}{y(1-ay)}.dy$
$ln(x+a)=\int \frac{1}{y}+\frac{a}{1-ay}.dy$
$ln(x+a)=ln\left(y\right)-ln(1-ay)+lnc$
$ln(x+a)=ln\left(\frac{cy}{1-ay}\right)$
$(x+a)=\frac{cy}{1-ay}$.