Question

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

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

Answer 1

The correct option is A $y=k(1-ay)(x+a)$

$(y-x\frac{dy}{dx})=a(y^2+\frac{dy}{dx})$
$\Rightarrow (y-a{y}^2)=(a+x)\frac{dy}{dx}$
$\Rightarrow \int \frac{dy}{y(1-ay)}=\int \frac{dx}{a+x}$
$\Rightarrow \int [\frac{1}{y}+\frac{a}{1-ay}]dy=\ln (a+x)+\ln k$
$\Rightarrow \ln y-\ln (1-ay)=\ln \left[\right(a+x\left)k\right]$
$\Rightarrow \ln \left[\frac{y}{1-ay}\right]=\ln \left[k\right(x+a\left)\right]$
$\Rightarrow y=k(x+a)(1-ay)$