DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

$\text{y}=\sin{\text{x}} ...(1)$

Differentiating both sides of (1) with respect to x, we get

$\frac{\text{dy}}{\text{dx}}=\cos\text{x} ...(2)$

Differentiating both sides of (2) with respect to x, we get

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

$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=-\text{y}$ [Using (1)]

$\Rightarrow \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\text{y}=0$

it is the given differential equation.

Here, $\text{y}=\sin\text{x}$ satisfies the given differential equation; hence, it is a solution.

Also, when $\text{x}=0,\text{y}=\sin0=0,\text{i.e.},\text{y}(0)=0.$

And, when $\text{x}=0,\text{y}'=\cos 0=1,\text{i.e.,}\text{y}'(0)=1.$

Hence, $\text{y}=\sin\text{x}$ is the solution to the given initial value problem.