Question

Solution of $y-x\frac{dy}{dx}=5(y^2+\frac{dy}{dx})$, is:

• A. $(5+x)(1-5y)=cy$
• B. $\frac{5+x}{1-5y}=cy$
• C. $(5-x)(1+5y)=cy$
• D. $(5+x)(5-x)=c$

Answer 1

The correct option is A $(5+x)(1-5y)=cy$

$5y^2-y=-\frac{dy}{dx}(5+x)$

$\Rightarrow -\frac{dx}{5+x}=\frac{dy}{y(5y-1)}$
$\Rightarrow -\frac{dx}{5+x}=\frac{5y-(5y-1)}{y(5y-1)}dy$

Integrating both sides, we get
$\Rightarrow \int \frac{dx}{5+x}=\int (\frac{5}{1-5y}+\frac{1}{y})dy$

$\Rightarrow \log (5+x)=-\log (1-5y)+\log y+\log c;(\log c$ is a constant$)$

$\Rightarrow yc=(5+x)(1-5y)$