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Q.
The solution of the differential equation $\frac{d^{2} y}{d x^{2}}=\sin 3 x+e^{x}$ $+x^{2}$ when $y^{\prime}(0)=1$ and $y(0)=0$ is
Differential Equations
Solution:
Integrating the given differential equation, we have
$\frac{d y}{d x}=\frac{-\cos 3 x}{3}+e^{x}+\frac{x^{3}}{3}+C_{1}$ but $y'(0)=1$
so $1=\left(-\frac{1}{3}\right)+1+C_{1}$
$\Rightarrow C_{1}=1 / 3$.
Again integrating, we get
$y=\frac{-\sin 3 x}{9}+e^{x}+\frac{x^{4}}{12}+\frac{1}{3} x+C_{2}$
but $y(0)=0$ so $0=0+1+C_{2}$
$\Rightarrow C_{2}=-1$. Thus
$y=\frac{-\sin 3 x}{9}+e^{x}+\frac{x^{4}}{12}+\frac{1}{3} x-1$