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

Integrating the given differential equation, we have

$\frac{dy}{dx}=\frac{-\cos 3x}{3}+e^x+\frac{x^3}{3}+C_1$

But $y_1\left(0\right)=1$

So, $1=\left(-\frac{1}{3}\right)+1+C_1$

$\Rightarrow C_1=\frac{1}{3}$

$\therefore \frac{dy}{dx}=\frac{-\cos 3x}{3}+e^x+\frac{x^3}{3}+\frac{1}{3}$

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, $y=-\frac{\sin 3x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3}x-1$

Hence, option (a) correct.