Question

Solve:

$\frac{d^{2}y}{d{x}^2}=x-\sin x$ with $\frac{dy}{dx}=1$ and $y=1$ when $x=0$.

Answer 1

Given the differential equation,

$\frac{d^{2}y}{d{x}^2}=x-\sin x$.

Integrating we have,

$\frac{dy}{dx}=\frac{x^2}{2}+\cos x+c$ [ Where $c$ is integrating constant].......(1)

Given, $x=0$ then $\frac{dy}{dx}=1$ then,

$1=0+1+c$

or, $c=0$.

Then form (1) we get,

$\frac{dy}{dx}=\frac{x^2}{2}+\cos x$

Now, integrating we get,

$y=\frac{x^3}{6}+\sin x+c_1$ [ Where $c_1$ is integrating constant]

Given $y=1$ when $x=0$

or, $1=0+c_1$

or, $c_1=1$

So the solution is, $y=\frac{x^3}{6}+\sin x+1$.