Question

If $y\left(x\right)$ is the solution of the differential equation $(x+2)\frac{dy}{dx}=x^2+4x-9,x\ne -2$ and $y\left(0\right)=0$, then $y(-4)$ is equal to:

• A. $1$
• B. $0$
• C. $2$
• D. $-1$

Answer 1

The correct option is B $0$

$(x+2)\frac{dy}{dx}=x^2+4x-9$

$(x+2)\frac{dy}{dx}=(x+2{)}^2-13$

$\frac{dy}{dx}=(x+2)-13.\frac{1}{x+2}$

$dy=\left(\right(x+2)-13.\frac{1}{x+2})dx$

Integrating we get,

$y=\frac{(x+2{)}^2}{2}-13.\log |x+2|+c$
Given $y\left(0\right)=0\Rightarrow 0=2-13.\log 2+c\Rightarrow c=13\log 2-2$

$\therefore y\left(x\right)=\frac{(x+2{)}^2}{2}-13.\log |x+2|+13\log 2-2$

Hence $y(-4)=\frac{(-4+2{)}^2}{2}-13.\log |-4+2|+13\log 2-2=0$