Question

Let $y\left(x\right)$ is the solution of the differential equation $\left(x+2\right)\frac{dy}{dx}-\left(x+1\right)y=2$ . If $y\left(0\right)=-1$ , then the value of $y\left(2\right)$ is equal to

• A. $e^2+\frac{1}{2}$
• B. $e^2-\frac{1}{2}$
• C. $\frac{1}{2}-e^2$
• D. $e^2$

Answer 1

Answer: (B)

Given differential equation is $\frac{dy}{dx}-\frac{x+1}{x+2}y=\frac{2}{x+2}$
I.F. $=e^{-\int \frac{x+1}{x+2}dx}=e^{-x}\left(x+2\right)$
So the solution of the equation is $y\left(x+2\right)e^{-x}=-2e^{-x}+C$
Since $y\left(0\right)=1,$ we have $C=4$
So $y\left(2\right)=e^2-\frac{1}{2}$