2.33   ODE No. 33

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

$$-\frac{y(x{)}^2{f}^{\prime }(x)}{g(x)}+\frac{g^{\prime }(x)}{f(x)}+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 37.5997 (sec), leaf count = 114

$$\text{Solve}[c_1={\int }_1^{y(x)}(\frac{1}{(f(x)K[2]+g(x){)}^2}-{\int }_1^x-\frac{2(K[2]f^{\prime }(K[1])+g^{\prime }(K[1]))}{(K[2]f(K[1])+g(K[1]){)}^3}dK[1])dK[2]+{\int }_1^x\frac{g(K[1])g^{\prime }(K[1])-y(x{)}^{2}f(K[1])f^{\prime }(K[1])}{f(K[1])g(K[1])(y(x)f(K[1])+g(K[1]){)}^2}dK[1],y(x)]$$

✓ Maple : cpu = 0.734 (sec), leaf count = 58

$$\{y\left(x\right)=\frac{1}{{\left(f\left(x\right)\right)}^2}(-g\left(x\right)f\left(x\right)\int \frac{\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)}{g\left(x\right){\left(f\left(x\right)\right)}^2}\mathrm{d}x-f\left(x\right)g\left(x\right)\mathit{\_}\mathit{C}\mathit{1}-1){(\int \frac{\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)}{g\left(x\right){\left(f\left(x\right)\right)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})}^{-1}\}$$

Hand solution

$$\begin{array}{rl}-\frac{f^{\prime }}{g}{y}^2+\frac{g^{\prime }}{f}+y^{\prime }& =0\\ y^{\prime }& =-\frac{g^{\prime }}{f}+\frac{f^{\prime }}{g}{y}^2\\ \text{(1)}& & =P\left(x\right)+Q\left(x\right)y+R\left(x\right)y^2\end{array}$$

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.