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

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

Answers (1)

Answer:

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}

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