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  • Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 52 maths

Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 52 maths

Answers (1)

Answer: 2 y=e^{x}(\sin x-\cos x)+1

Hint: Separate the terms of x and y and then integrate them.

Given: Curve passing through (0,0) with differential equation

\frac{d y}{d x}=e^{x} \sin x

Solution:

        \begin{aligned} &\frac{d y}{d x}=e^{x} \sin x \\\\ &\Rightarrow d y=e^{x} \sin x d x \end{aligned}

          Integrating both sides

        \int d y=\int e^{x} \sin x d x \Rightarrow y=\int e^{x} \sin x d x            .................(*)

        \begin{aligned} &y=\sin x e^{x}-\int \cos x e^{x} d x \\\\ &\Rightarrow y=\sin x e^{x}-\left[\cos x e^{x}+\int \sin x e^{x} d x\right]+c \\\\ &\Rightarrow y=\sin x e^{x}-\cos x e^{x}-y+c \\\\ &\Rightarrow 2 y=e^{x}(\sin x-\cos x)+c \end{aligned}..............(1)

        The curve passes through (0,0)           [∴ of given]

        \begin{aligned} &0=e^{0}(\sin 0-\cos 0)+c\\\\ &\Rightarrow 0=1(0-1)+c \Rightarrow c=1\\\\ &\operatorname{Put\; in}(1)\\\\ &2 y=e^{x}(\sin x-\cos x)+1 \end{aligned}

        

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