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  • Provide Solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 21

Provide Solution for RD Sharma Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 21

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Answer:   y=x^{4}+x^{2} \log x+C x^{2}

Hint: To solve this we convert   formula.

Give: x d y=\left(2 y+2 x^{4}+x^{2}\right) d x \\

Solution:   \begin{aligned} & & \frac{d y}{d x}=\frac{2 y+2 x^{4}+x^{2}}{x} \end{aligned}

\begin{aligned} &=\frac{d y}{d x}=\frac{2 y}{x}+2 x^{3}+x \\ &=\frac{d y}{d x}-\frac{2 y}{x}=2 x^{3}+x \\ &\frac{d y}{d x}+P y=Q \end{aligned}
\begin{aligned} &P=-\frac{2}{x}, Q=2 x^{3}+x \\ &I f=e^{\int P d x} \\ &=e^{-\int \frac{2}{x} d x} \\ &=e^{=2 \log x} \\ &=e^{\log x^{-2}} \\ &=x^{-2} \\ &=\frac{1}{x^{2}} \quad\left[e^{\log x}=x\right] \end{aligned}

\begin{aligned} &=y I f=\int Q I f d x+C \\ &=\frac{y}{x^{2}}=\int \frac{2 x^{3}+x}{x^{2}} d x+C \\ &=\frac{y}{x^{2}}=\int 2 x+\frac{1}{x} d x+C \\ &=\frac{y}{x^{2}}=x^{2}+\log x+C \\ &=y=x^{4}+x^{2} \log x+C x^{2} \end{aligned}

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