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  • Explain solution RD Sharma class 12 chapter Differential Equation exercise 21.3 question 16 maths

Explain solution RD Sharma class 12 chapter Differential Equation exercise 21.3 question 16 maths

Answers (1)

Answer:

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

Hint:

Differentiate with respect to x

Use formula \frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^{2}}

Given:

y=c e^{\tan ^{-1} x}


Solution:

Differentiating on both sides with respect to x

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

Differentiating equation (i) with respect to x

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

\frac{d^{2} y}{d x^{2}}=c\left[\frac{\left(1+x^{2}\right) \frac{d}{d x}\left(e^{\tan ^{-1} x}\right)-e^{\tan ^{-1} x} \frac{d}{d x}\left(1+x^{2}\right)}{\left(1+x^{2}\right)^{2}}\right]

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

\frac{d^{2} y}{d x^{2}}=\left[\frac{c(1-2 x) e^{\tan ^{-1} x}}{\left(1+x^{2}\right)^{2}}\right]                                .............(ii)

Put (i) and (ii) in given equation

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

\begin{aligned} &=\left(1+x^{2}\right) \frac{c(1-2 x) e^{\tan ^{-1} x}}{\left(1+x^{2}\right)^{2}}+(2 x-1) \frac{c e^{\tan ^{-1} x}}{\left(1+x^{2}\right)} \\\\ &=\frac{c(1-2 x) e^{\tan ^{-1} x}}{\left(1+x^{2}\right)}-(1-2 x) \frac{c e^{\tan ^{-1} x}}{\left(1+x^{2}\right)} \\\\ &=0 \\\\ &=R H S \end{aligned}

Thus, y=c e^{\tan ^{-1} x} is a solution of the differential equation.

 

Posted by

infoexpert26

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