2.96   ODE No. 96

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

$$x{y}^{\prime }(x)-y(x{)}^2+1=0$$ ✓ Mathematica : cpu = 0.057777 (sec), leaf count = 33

$$\left\{\{y(x)\to \frac{1-e^{2c_1}{x}^2}{1+e^{2c_1}{x}^2}\}\right\}$$ ✓ Maple : cpu = 0.034 (sec), leaf count = 11

$$\{y\left(x\right)=-\tanh (\ln \left(x\right)+\mathit{\_}\mathit{C}\mathit{1})\}$$

Hand solution

$$x{y}^{\prime }-y^2+1=0$$

This is Riccati first order non-linear. But it is separable. Hence$$\begin{array}{}\text{(1)}& y^{\prime }=\frac{y^2-1}{x}\end{array}$$

Hence

$$\begin{array}{rl}\frac{dy}{dx}& =\frac{y^2-1}{x}\\ \frac{dy}{y^2-1}& =\frac{dx}{x}\end{array}$$

Integrating

$$\begin{array}{rl}-{\tanh }^{-1}\left(y\right)& =\ln x+C\\ y& =-\tanh (\ln x+C)\end{array}$$

Verification

restart; 
ode:=x*diff(y(x),x)-y(x)^2+1=0; 
my_sol:=-tanh(ln(x)+_C1); 
odetest(y(x)=my_sol,ode); 
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