Question

The solution of the differential equation $\frac{d^{2}y}{d{x}^2}=\sin 3x+e^x+x^2$ when $y_1\left(0\right)=1$ and $y\left(0\right)=0$ is

• A. $\frac{-\sin 3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x-1$
• B. $\frac{-\sin 3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x$
• C. $\frac{-\cos 3x}{3}+e^x+\frac{x^4}{12}+\frac{1}{3}x+1$
• D. none of these

Answer 1

The correct option is A $\frac{-\sin 3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x-1$

Integrating the given differential equation, we have
$\frac{dy}{dx}=\frac{-cos3x}{3}+e^x+\frac{x^3}{3}+C_1$
but $y_1\left(0\right)=1$ so $1=(-1/3)+1+C_1\Rightarrow C_1=1/3$.
Again integrating, we get
$y=\frac{-\sin 3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x+C_2$
but $y\left(0\right)=0$ so $0=0+1+C_2\Rightarrow C_2=-1$.Thus
$y=\frac{-\sin 3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x-1$