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  • Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 45 sub question (viii) maths

Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 45 sub question (viii) maths

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Answer: \tan ^{-1} y=\frac{x^{2}}{2}+x

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

Given: \frac{d y}{d x}=1+x+y^{2}+x y^{2}, \text { when } y=0, x=0

Solution:

        \begin{aligned} &\frac{d y}{d x}=1+x+y^{2}+x y^{2} \\\\ &\frac{d y}{d x}=(1+x)+y^{2}(1+x)=(1+x)\left(1+y^{2}\right) \\\\ &\Rightarrow \frac{d y}{1+y^{2}}=(1+x) d x \end{aligned}

        Integrating both sides

        \begin{aligned} &\int \frac{1}{1+y^{2}} d y=\int(1+x) d x \\\\ &\Rightarrow \tan ^{-1} y=x+\frac{x^{2}}{2}+c \end{aligned}        ..............(1)

        Given that when y=0, x=0

        \therefore \tan ^{-1}(0)=0+\frac{0}{2}+c \Rightarrow c=0

        Put in (1) we get

        \begin{aligned} &\tan ^{-1} y=x+\frac{x^{2}}{2}+0 \\\\ &\Rightarrow \tan ^{-1} y=x+\frac{x^{2}}{2} \end{aligned}

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