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Need solution for RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise 21.10 Question 11 textbook solution.

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Answer : 5 x y=e^{5}+C

Hint: To solve this equation we use e\int ^{pdx}  formula.

Give : x \frac{d y}{d x}+\frac{y}{x}=x^{3}

Solution : \frac{d y}{d x}+P y=Q

                \begin{aligned} &P=\frac{1}{x}, Q=x^{3} \\ \end{aligned}

                \begin{aligned} &I f=e^{\int P d x} \\ \end{aligned}

                 \begin{aligned} &=e^{\int \frac{1}{x} d x} \\ \end{aligned}

                 \begin{aligned} &=e^{\log x} \\ \end{aligned}

                 \begin{aligned} &=x \end{aligned}

                \begin{aligned} &y I f=\int Q I f+C \\ \end{aligned}

                \begin{aligned} &=y x=\int x^{3} x d x+C \\ \end{aligned}

              \begin{aligned} &=y x=\int x^{4} d x+C \; \; \; \; \; \; \; \; \; \; \quad\left[\int x^{n} d x=\frac{x^{n+1}}{n+1}\right] \\ \end{aligned}

              \begin{aligned} &=x y=\frac{x^{5}}{5}+C \\ \end{aligned}

               \begin{aligned} &=y=\frac{x^{4}}{5}+\frac{C}{x} \end{aligned}

                =5 x y=x^{5}+C

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