#### Please solve RD Sharma class 12 chapter Differential Equations exercise 21.7 question 37 sub question (i) maths textbook solution

Answer: $y^{2}+2=\frac{c^{1}}{x^{2}+2}$

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

Given: $\left(x y^{2}+2 x\right) d x+\left(x y^{2}+2 y\right) d y=0$

Solution: $\left(x y^{2}+2 x\right) d x+\left(x y^{2}+2 y\right) d y=0$

\begin{aligned} &x\left(y^{2}+2\right) d x+y\left(x^{2}+2\right) d y=0 \\\\ &y\left(x^{2}+2\right) d y=-x\left(y^{2}+2\right) d x \\\\ &\frac{y}{\left(y^{2}+2\right)} d y=\frac{-x}{\left(x^{2}+2\right)} d x \end{aligned}

Integrating both sides

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

\begin{aligned} &\Rightarrow \frac{1}{2} \log \left|y^{2}+2\right|=-\frac{1}{2} \log \left|x^{2}+2\right|+\log c \\\\ &\Rightarrow \frac{1}{2}\left[\log \left|y^{2}+2\right|+\log \left|x^{2}+2\right|\right]=\log c \end{aligned}

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