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  • Please solve RD Sharma class 12 chapter Differential Equation exercise 21.3 question 17 maths textbook solution

Please solve RD Sharma class 12 chapter Differential Equation exercise 21.3 question 17 maths textbook solution

Answers (1)

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Answer:

y=e^{m \cos ^{-1} x}  is a solution of differential equation

Hint:

Differentiate with respect to x and put values in given equation.

Given:

y=e^{m \cos ^{-1} x} 

Solution:

Differentiating on both sides with respect to x

\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x}\left(e^{m \cos ^{-1} x}\right) \\\\ &\frac{d y}{d x}=m e^{m \cos ^{-1} x}\left(\frac{-1}{\sqrt{1+x^{2}}}\right) \\\\ &\frac{d y}{d x}=\left(\frac{-m e^{m \cos ^{-1} x}}{\sqrt{1+x^{2}}}\right) \end{aligned}                        ............(i)

Differentiating equation (i) with respect to x,

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{-m\left[\sqrt{1+x^{2}} \frac{d}{d x}\left(e^{m \cos ^{-1} x}\right)-e^{m \cos ^{-1} x} \frac{d}{d x}\left(\sqrt{1+x^{2}}\right)\right]}{\left(\sqrt{1+x^{2}}\right)^{2}} \\\\ &\frac{d^{2} y}{d x^{2}}=\frac{-m\left[\sqrt{1+x^{2}}\left(\frac{m e^{m \cos ^{-1} x}}{\sqrt{1+x^{2}}}\right)-e^{m \cos ^{-1} x}\left(\frac{-x}{\sqrt{1+x^{2}}}\right)\right]}{\left(\sqrt{1+x^{2}}\right)^{2}} \end{aligned}

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{-m\left[-m e^{m \cos ^{-1} x}+\left(\frac{x e^{m \cos ^{-1} x}}{\sqrt{1+x^{2}}}\right)\right]}{\left(\sqrt{1+x^{2}}\right)^{2}} \\\\ &\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}=m^{2} e^{m \cos ^{-1} x}+\left(\frac{-m x e^{m \cos ^{-1} x}}{\sqrt{1+x^{2}}}\right) \end{aligned}

\begin{aligned} &\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}=m^{2} y+x \frac{d y}{d x} \\\\ &\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-m^{2} y=0 \end{aligned}

Hence proved, the given function is the solution to given differential equation

 

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