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Please solve RD Sharma class 12 chapter Differential Equation exercise 21.3 question 13 maths textbook solution

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Answer:

y=c x+2 c^{2}  is a solution of differential equation

Hint:

Just differentiate once and put value in given differential problem

Given:

y=c x+2 c^{2}  is a solution of the equation


Solution:

Differentiating on both sides with respect to x

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

Put value in given problem as follows

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

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

Thus, y=c x+2 c^{2}  is a solution of differential equation.

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