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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 23 Maths Textbook Solution.

Provide Solution For R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise  Revision Exercise Question 23 Maths Textbook Solution.

Answers (1)

Answer:y=log|logx|+\frac{x^{2}}{2}-\frac{x\sin 2x}{4}-\frac{\cos 2x}{8}+c

Hint:  integrate both sides and then apply the formula of

Given: \frac{dy}{dx}-x\sin ^{2}x=\frac{1}{xlogx}

Solution: \frac{dy}{dx}-x\sin ^{2}x=\frac{1}{xlogx}

          \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{x} \log x}+\mathrm{x} \sin ^{2} \mathrm{x}

         \Rightarrow \frac{\mathrm{d} y}{\mathrm{dx}}=\frac{1}{\mathrm{x} \operatorname{logx}}+\frac{\mathrm{x}}{2}(1-\cos 2 \mathrm{x})\left[\because \operatorname{Cos} 2 \mathrm{x}=1-2 \sin ^{2} \mathrm{x}\right]

        \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{x} \log x}+\frac{\mathrm{x}}{2}-\frac{x \operatorname{Cos} 2 \mathrm{x}}{2} \\

        \Rightarrow \int \mathrm{dy}=\int\left(\frac{1}{\mathrm{xlog} \mathrm{x}}+\frac{\mathrm{x}}{2}-\frac{x \operatorname{Cos} 2 \mathrm{x}}{2}\right) \mathrm{d} \mathrm{x} \quad \text { (integrate both sides) } \\

        \Rightarrow \int \mathrm{dy}=\int\left(\frac{1}{\mathrm{x} \log \mathrm{x}} \mathrm{dx}+\frac{1}{2} \int \mathrm{x} \mathrm{dx}-\frac{1}{2}(\mathrm{x} \operatorname{Cos} 2 \mathrm{x}) \mathrm{d} \mathrm{x}\right. \\

        \Rightarrow \mathrm{y}=\log |\log \mathrm{x}|+\frac{\mathrm{x}^{2}}{2}-\frac{\mathrm{x}}{2} \int \cos 2 \mathrm{x} \mathrm{dx}+\frac{1}{2} \int \frac{\sin 2 \mathrm{x}}{2} \mathrm{dx} \\

       \Rightarrow \mathrm{y}=\log |\log \mathrm{x}|+\frac{\mathrm{x}^{2}}{2}-\frac{\mathrm{x}}{2} \cdot \frac{\sin 2 \mathrm{x}}{2}+\frac{1}{2}\left(-\frac{\cos 2 \mathrm{x}}{4}\right)+\mathrm{C} \\

       \Rightarrow \mathrm{y}=\log |\log \mathrm{x}|+\frac{\mathrm{x}^{2}}{2}-\frac{\mathrm{x} \sin 2 \mathrm{x}}{4}-\frac{\cos 2 \mathrm{x}}{8}+\mathrm{C}

 

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