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  • Please Solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 25 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 25 Maths Textbook Solution.

Answers (1)

best_answer

Answer:  x=\tan y+C \sqrt{\tan y}

Hint: To solve this equation we use \tan x  and convert it to  \sin x  and  \cos x

Give:  \begin{aligned} &(x+\tan y) d y=\sin 2 y d x \\ & \end{aligned}

Solution: (x+\tan y) d y=\sin 2 y d x

\begin{aligned} &=\frac{d x}{d y}=\frac{(x+\tan y)}{\sin 2 y} \\ &=\frac{d x}{d y}=\frac{x}{\sin 2 y}+\frac{\tan y}{\sin 2 y} \\ &=\frac{d x}{d y}=x \operatorname{cosec} 2 y+\frac{\frac{\sin y}{\cos y}}{2 \sin y \cos y} \\ &=\frac{d x}{d y}=x \operatorname{cosec} 2 y+\frac{\frac{\sin y}{\cos y}}{2 \sin y \cos y} \end{aligned}

\begin{aligned} &=\frac{d x}{d y}=x \operatorname{cosec} 2 y+\frac{1}{2 \cos ^{2} y} \\ &=\frac{d x}{d y}=x \operatorname{cosec} 2 y+\frac{1}{2} \sec ^{2} y \\ &=\frac{d x}{d y}+R x=S \end{aligned}

\begin{aligned} &R=\operatorname{cosec} 2 y, S=\frac{1}{2} \sec ^{2} y \\ & \end{aligned}

\text { If }=e^{\int R d y} \\

=e^{\int(-\operatorname{cosec} 2 y) d y} \\

=e^{-\log |\operatorname{cosec} 2 y-\cot 2 y|} \\

=\operatorname{cosec} 2 y-\cot 2 y \\

=\frac{1}{\sin 2 y}-\frac{\cos 2 y}{\sin 2 y} \\

=\frac{1-\cos 2 y}{\sin 2 y}

\begin{aligned} &=\frac{2 \sin ^{2} y}{2 \sin y \cos y} \\ & \end{aligned}

=\frac{\sin y}{\cos y} \\

=\tan y \\

=e^{-\log |\tan y|} \\

=e^{\log |\cot y|} \\

=\cot y

\begin{aligned} &=x I f=\int S I f+C \\ &=x \cot y=\int \frac{1}{2} \sec ^{2} y \cot y d y+C \\ &=x \cot y=\int \frac{1}{2 \cos ^{2} y} \frac{\cos y}{\sin y} d y+C \\ &=x \cot y=\int \frac{1}{2 \cos y} \frac{1}{\sin y} d y+C \\ &=x \cot y=\int \frac{1}{\sin 2 y} d y+C \\ &=x \cot y=\frac{1}{2} \log |\operatorname{cosec} 2 y-\cot 2 y|+C \end{aligned}

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