3.3.11.1 Examples

3.3.11.1.1 Example 1 Solve

$$y^{\prime }=-\frac{y(y^2+3x^2+2x)}{x^2+y^2}$$

Applying change of variables $y=ux$ results in

$$u^{\prime }=-\frac{u(u^2+3)}{u^2+1}\frac{x+1}{x}$$

Which is separable. Solving this for $u\left(x\right)$ by integration gives

$$\begin{array}{rl}\int \frac{1}{-\frac{u(u^2+3)}{u^2+1}}du& =\int \frac{x+1}{x}dx-\frac{u(u^2+3)}{u^2+1}\ne 0\\ \frac{1}{3}\ln \left((u^2+3)u\right)+x+\ln \left(x\right)& =c_1\end{array}$$

Hence the solution  in $y\left(x\right)$ is

$$\frac{1}{3}\ln \left(({\left(\frac{y}{x}\right)}^2+3)\frac{y}{x}\right)+x+\ln \left(x\right)=c_1$$

Singular solution is when $u(u^2+3)=0$ or $u=0,u=\pm i\sqrt{3}$ which implies $y=0$ and $y=\pm i\sqrt{3}x$. Hence the solutions are

$$\begin{array}{rl}\frac{1}{3}\ln \left(({\left(\frac{y}{x}\right)}^2+3)\frac{y}{x}\right)+x+\ln \left(x\right)& =c_1\\ y& =0\\ y& =i\sqrt{3}x\\ y& =-i\sqrt{3}x\end{array}$$