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  • Provide solution for RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Revision Exercise (RE) Question 55 textbook solution.

Provide solution for  RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Revision Exercise (RE) Question 55 textbook solution.

Answers (1)

Answer :  y e^{\tan x}=e^{\tan x}(\tan x-1)+c

Hint       : You must know the rules of solving differential equation and integration

Given    : \cos ^{2} x \frac{d y}{d x}+y=\tan x

Solution : \cos ^{2} x \frac{d y}{d x}+y=\tan x

             \begin{aligned} \frac{d y}{d x}+\frac{y}{\cos ^{2} x} &=\frac{\tan x}{\cos ^{2} x} \\ \frac{d y}{d x}+\left(\sec ^{2} x\right) y &=(\tan x) \sec ^{2} x \end{aligned}

Comparing with ,

            \frac{dy}{dx}+Py=Q

Where, \mathrm{P}=\sec ^{2} x, \mathrm{Q}=(\tan x)\left(\sec ^{2} x\right)

Now,

\begin{aligned} \text { I.F } &=e^{\int \sec ^{2} x d x} \\ &=e^{\tan x} \end{aligned}

So, the solution is

\begin{aligned} y \times I . F &=\int(I . F \times Q) d x+c \\ y \times e^{\tan x} &=\int e^{\tan x} \times(\tan x)\left(\sec ^{2} x\right) d x+c \\ y e^{\tan x} &=I+C \end{aligned}

Now,

I=\int \tan x\left(\sec ^{2} x\right) \times e^{\tan x} d x

Integrating by parts,

\begin{aligned} &\text { Put } t=\tan x \\ &d t=\sec ^{2} x d x \end{aligned}

Therefore,

\begin{aligned} I &=\int\left(t \times e^{t}\right) d t \\ I &=t \times \int e^{t} d t-\int\left(\frac{d t}{d t} \times \int e^{t} d t\right) d t \\ &=t e^{t}-\int e^{t} d t \\ &=t e^{t}-e^{t} \end{aligned}

Therefore,

\begin{aligned} I &=\tan x e^{\tan x}-e^{\tan x} \\ &=e^{\tan x}(\tan x-1) \end{aligned}

Hence, the required solution is

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

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