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Need solution for RD Sharma maths class 12 chapter Differential Equations exercise 21.7 question 45 sub question (iii)

Answers (1)


Answer: y e^{2 x}+1=0

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

Given: \frac{d y}{d x}=2 e^{2 x} y^{2}, y(0)=-1


        \begin{aligned} &\frac{d y}{d x}=2 e^{2 x} y^{2} \\\\ &\Rightarrow \frac{d y}{y^{2}}=2 e^{2 x} d x \end{aligned}

        Integrating both sides

        \begin{aligned} &\int \frac{1}{y^{2}} d y=\int 2 e^{2 x} d x \\\\ &\Rightarrow \int y^{-2} d y=\int 2 e^{2 x} d x \end{aligned}

        \Rightarrow \frac{y^{-2+1}}{-2+1}=\frac{2 e^{2 x}}{2}+c \Rightarrow \frac{-1}{y}=e^{2 x}+c        ..............(1)

        Given that  y(0)=-1 \text { i.e at } x=0 ; y=-1

        \begin{aligned} &\Rightarrow \frac{-1}{-1}=e^{2(0)}+c \Rightarrow 1=e^{0}+c \Rightarrow 1=1+c \Rightarrow c=0 \\\\ &{\left[\therefore e^{0}=1\right]} \end{aligned}

        Put in (1)

        \Rightarrow \frac{-1}{y}=e^{2 x}+0 \Rightarrow y e^{2 x}+1=0


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