Question

Find general solution of $\frac{2dy}{dx}=\frac{y(x+1)}{x}$.

• A. $\log y^2=x+\log |x|+k$
• B. $\log y^2=x+\log \left(x\right)+k$
• C. $\log y=x+\log \left(x\right)+k$
• D. $2\log y=x-\log |x|+k$

Answer 1

The correct option is A $\log y^2=x+\log |x|+k$

Given,

$2\frac{dy}{dx}=\frac{y(x+1)}{x}$

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

Integrating both sides w.r.t $x$

$\int \frac{2dy}{y}=\int \frac{x+1}{x}dx$

$\int \frac{2dy}{y}=\int 1+\frac{1}{x}dx$

$\Rightarrow 2\log y=x+\log |x|+c$

$\Rightarrow \log y^2=x+\log |x|+c$