#### Need solution for RD Sharma maths class 12 chapter 21 Differential Equation exercise Fill in the blank question 31

$\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+y=0$

Hint:

Use simple differentiation w.r.t x

Given:

The differential equation for which y=a cos x + b sin x is a solution, is ____

Solution:

$y=a\, cos\, x+b\, sin\, x \qquad \qquad \dots(i)$

differentiating both side w.r.t x, we get

\begin{aligned} &\frac{\mathrm{d} y}{\mathrm{d} x}=a\left ( \frac{dcos\, x}{dx} \right )+b\left ( \frac{dsin\, x}{dx} \right ) \\ &=a(-sin\, x)+b(cos\, x) \\ &=-asin\, x+bcos\, x \end{aligned}

Again differentiating both side w.r.t x, we get

Now, we have to verify

\begin{aligned} &\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+y=0 \end{aligned}

Taking L.H.S

\begin{aligned} &\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=(-acos\, x-bsin\, x) \\ &\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=-y \qquad \qquad [using(i)] \\ &\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+y=0 \end{aligned}

∴So, the differential equation is

$\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+y=0$