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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 22 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 22 Maths Textbook Solution.

Answers (1)

Answer: y=\left ( x^{2}-2x +2\right )e^{x}+c

Hint: You must know about the formula of \int uv\; \; dx

Given:\frac{dy}{dx}=x^{2}e^{x}

Solution:\frac{dy}{dx}=x^{2}e^{x}

\begin{aligned} &\Rightarrow d y=x^{2} e^{x} d x\\ &\Rightarrow \int d y=\int x^{2} e^{x} d x \text { (integrate both sides) }\\ &\Rightarrow \int d y=x^{2} \int e^{x} d x-\int\left(2 x \int e^{x} d x\right) d x\\ &\left[\because \int u v d x=u \int v d x-\int\left(\frac{d}{d x} u \int v d x\right) d x\right]\\ &\Rightarrow y=x^{2} e^{x}-2 \int x e^{x} d x\\ &\Rightarrow y=x^{2} e^{x}-2 x \int e^{x} d x+2 \int e^{x} d x\\ &\left[\because \int \mathrm{uv} \mathrm{d} \mathrm{x}=\mathrm{u} \int \mathrm{v} \mathrm{d} \mathrm{x}-\int\left(\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{u} \int \mathrm{v} \mathrm{d} \mathrm{x}\right) \mathrm{dx}\right]\\ &\Rightarrow y=x^{2} e^{x}-2 x e^{x}+2 e^{x}+C\\ &\Rightarrow y=\left(x^{2}-2 x+2\right) e^{x}+C \end{aligned}

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