2.33   ODE No. 33

\[ -\frac {y(x)^2 f'(x)}{g(x)}+\frac {g'(x)}{f(x)}+y'(x)=0 \] Mathematica : cpu = 0.402088 (sec), leaf count = 160


\[\text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{(g(x)+f(x) K[2])^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2 f'(K[1])-g(K[1]) g'(K[1])\right )}{g(K[1]) (g(K[1])+f(K[1]) K[2])^3}-\frac {2 K[2] f'(K[1])}{g(K[1]) (g(K[1])+f(K[1]) K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2 f'(K[1])-g(K[1]) g'(K[1])}{f(K[1]) g(K[1]) (g(K[1])+f(K[1]) y(x))^2}dK[1]=c_1,y(x)\right ]\] Maple : cpu = 0.427 (sec), leaf count = 58


\[y \relax (x ) = \frac {-g \relax (x ) f \relax (x ) \left (\int \frac {\frac {d}{d x}f \relax (x )}{g \relax (x ) f \relax (x )^{2}}d x \right )-g \relax (x ) f \relax (x ) c_{1}-1}{f \relax (x )^{2} \left (\int \frac {\frac {d}{d x}f \relax (x )}{g \relax (x ) f \relax (x )^{2}}d x +c_{1}\right )}\]

Hand solution

\begin {align} -\frac {f^{\prime }}{g}y^{2}+\frac {g^{\prime }}{f}+y^{\prime } & =0\nonumber \\ y^{\prime } & =-\frac {g^{\prime }}{f}+\frac {f^{\prime }}{g}y^{2}\nonumber \\ & =P\relax (x) +Q\relax (x) y+R\relax (x) y^{2}\tag {1} \end {align}

This is Ricatti first order non-linear ODE. \(P\relax (x) =-\frac {g^{\prime }}{f},Q\relax (x) =0,R\relax (x) =\frac {f^{\prime }}{g}\).

To do.