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

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

Answers (1)

Answer:

 \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

Posted by

Gurleen Kaur

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