Question

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

• A. 0
• B. 1
• C. -1
• D. 2

Answer 1

Answer: (A)

$\left(x+2\right)\frac{dy}{dx}=x^2+4x-9x\ne -2$
$\frac{dy}{dx}=\frac{x^2+4x-9}{x+2}$
$dy=\frac{x^2+4x-9}{x+2}dx$
$\int dx=\int \frac{x^2+4x-9}{x+2}dx$
$y=\int \left(x+2-\frac{13}{x+2}\right)dx$
$y=\int \left(x+2\right)dx-13\int \frac{1}{x+2}dx$
$y-\frac{x^2}{2}+2x-13log|x+2|+c$
Given that $y=\left(0\right)=0$
$0=-13\log 2+c$
$y=\frac{x^2}{2}+2x-13\log |x+2|+13\log 2$
$y\left(-4\right)=8-8-13\log 2+13\log$
$2=0$