#### Need solution for RD Sharma maths class 12 chapter Differential Equation exercise 21.3 question 11

$y=\frac{c-x}{(1+c x)}$  is a solution of differential equation

Hint:

Differentiate the given solution and substitute in the differential equation.

Given:

$y=\frac{c-x}{(1+c x)}$  is a solution of the equation

Solution:

Differentiating on both sides with respect to $x$

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

\begin{aligned} &\frac{d y}{d x}=\frac{[(1+c x)(0-1)]-[(c-x)(0+c)]}{(1+c x)^{2}} \\\\ &\frac{d y}{d x}=\frac{[(1+c x)(-1)]-[(c-x)(c)]}{(1+c x)^{2}} \end{aligned}

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

Put equation (i) in differential equation as follows

\begin{aligned} &\left(1+x^{2}\right) \frac{d y}{d x}+\left(1+y^{2}\right)=0 \\\\ &L H S=\left(1+x^{2}\right) \frac{d y}{d x}+\left(1+y^{2}\right) \end{aligned}

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

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

$=\left[\frac{-\left(1+x^{2}\right)\left(1+c^{2}\right)}{(1+c x)^{2}}\right]+\frac{1+2 c x+c^{2} x^{2}+c^{2}-2 c x+x^{2}}{(1+c x)^{2}}$

\begin{aligned} &=\left[\frac{-\left(1+x^{2}\right)\left(1+c^{2}\right)}{(1+c x)^{2}}\right]+\frac{1+c^{2} x^{2}+c^{2}+x^{2}}{(1+c x)^{2}} \\\\ &=\left[\frac{-\left(1+x^{2}\right)\left(1+c^{2}\right)}{(1+c x)^{2}}\right]+\left[\frac{\left(1+x^{2}\right)\left(1+c^{2}\right)}{(1+c x)^{2}}\right] \\\\ &=0 \\\\ &=R H S \end{aligned}