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)

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