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Explain solution RD Sharma class 12 Chapter 21 Differential Equation exercise 21.10 question 11

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Answer:  5 x y=e^{5}+C

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

Give:  \begin{aligned} &x \frac{d y}{d x}+\frac{y}{x}=x^{3} \\ & \end{aligned}

Solution:  \frac{d y}{d x}+P y=Q

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

I\! f=e^{\int P d x} \\

=e^{\int \frac{1}{x} d x} \\

=e^{\log x} \\

=x \\

y I \! f=\int Q I\! f+C

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

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

=x y=\frac{x^{5}}{5}+C \\


=5 x y=x^{5}+C

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