2.96   ODE No. 96

\[ x y'(x)-y(x)^2+1=0 \] Mathematica : cpu = 0.0694451 (sec), leaf count = 33


\[\left \{\left \{y(x)\to \frac {1-e^{2 c_1} x^2}{1+e^{2 c_1} x^2}\right \}\right \}\] Maple : cpu = 0.037 (sec), leaf count = 11


\[y \relax (x ) = -\tanh \left (\ln \relax (x )+c_{1}\right )\]

Hand solution

\[ xy^{\prime }-y^{2}+1=0 \]

This is Riccati first order non-linear. But it is separable. Hence\begin {equation} y^{\prime }=\frac {y^{2}-1}{x} \tag {1} \end {equation}

Hence

\begin {align*} \frac {dy}{dx} & =\frac {y^{2}-1}{x}\\ \frac {dy}{y^{2}-1} & =\frac {dx}{x} \end {align*}

Integrating

\begin {align*} -\tanh ^{-1}\relax (y) & =\ln x+C\\ y & =-\tanh \left (\ln x+C\right ) \end {align*}

Verification