2.101   ODE No. 101

1. Problem in Latex
2. Mathematica input
3. Maple input

$$x{y}^{\prime }(x)+xy(x{)}^2-y(x)=0$$ ✓ Mathematica : cpu = 0.0719045 (sec), leaf count = 18

$$\left\{\{y(x)\to \frac{2x}{x^2+2c_1}\}\right\}$$ ✓ Maple : cpu = 0.01 (sec), leaf count = 16

$$\{y\left(x\right)=2\frac{x}{x^2+2\mathit{\_}\mathit{C}\mathit{1}}\}$$

Hand solution

$$\begin{array}{rl}x{y}^{\prime }+x{y}^2-y& =0\\ \text{(1)}& y^{\prime }& =\frac{1}{x}y-y^2\end{array}$$

This is of the form $y^{\prime }=f_0+f_{1}y+f_2{y}^2$ with $f_0=0,f_1=\frac{1}{x},f_2=-1$. Since $f_0=0$ this is Bernoulli differential equation. We always start by dividing by $y^2$$$\frac{y^{\prime }}{y^2}=\frac{1}{x}\frac{1}{y}-1$$ Then $u=\frac{1}{y}$ or $y=\frac{1}{u}$, therefore $y^{\prime }=-\frac{u^{\prime }}{u^2}$. Equating this to RHS of (1) gives$$\begin{array}{rl}-\frac{u^{\prime }}{u^2}{u}^2& =\frac{1}{x}u-1\\ -u^{\prime }& =\frac{u}{x}-1\\ u^{\prime }+\frac{u}{x}& =1\end{array}$$

Integrating factor is $e^{\int \frac{1}{x}dx}=x$ and the above becomes$$d\left(xu\right)=x$$ Integrating$$\begin{array}{rl}xu& =\frac{x^2}{2}+C\\ u& =\frac{x}{2}+\frac{C}{x}\\ & =\frac{x^2+2C}{2x}\end{array}$$

Hence $$\begin{array}{rl}y& =\frac{1}{u}\\ & =\frac{2x}{x^2+2C}\end{array}$$

Verification

restart; 
ode:=x*diff(y(x),x)+x*y(x)^2-y=0; 
my_solution:=2*x/(x^2+2*_C1); 
odetest(y(x)=my_solution,ode); 
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