Chapter 10 - Introduction to Differential Equations - 10.1 Solving Differential Equations - Exercises - Page 505: 26

$$y=\frac{C_1e^{x^{2} / 2}}{\sqrt{1+x^{2}}}$$

Work Step by Step

Given $$\left(1+x^{2}\right) y'=x^{3} y$$ Then \begin{align*} \left(1+x^{2}\right) \frac{d y}{d x}&=x^{3} y\\ \frac{1}{y} d y&=\frac{x^{3}}{1+x^{2}} d x \end{align*} Hence \begin{aligned} \int \frac{1}{y} d y &=\int\left(x-\frac{x}{1+x^{2}}\right) d x \\ \ln y &=\frac{x^{2}}{2}-\frac{1}{2} \ln \left(1+x^{2}\right)+C \\ y &=\frac{C_1e^{x^{2} / 2}}{\sqrt{1+x^{2}}} \end{aligned}

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