2.33   ODE No. 33

$-\frac {y(x)^2 f'(x)}{g(x)}+\frac {g'(x)}{f(x)}+y'(x)=0$ Mathematica : cpu = 0.333181 (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 = 0.393 (sec), leaf count = 58

$\left \{ y \left ( x \right ) ={\frac {1}{ \left ( f \left ( x \right ) \right ) ^{2}} \left ( -g \left ( x \right ) f \left ( x \right ) \int \!{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{g \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x-g \left ( x \right ) f \left ( x \right ) {\it \_C1}-1 \right ) \left ( \int \!{\frac {{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) }{g \left ( x \right ) \left ( f \left ( x \right ) \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) ^{-1}} \right \}$

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\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}\tag {1} \end {align}

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

To do.