2.53   ODE No. 53

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

$$f(x{)}^{1-n}{g}^{\prime }(x)y(x{)}^n(-(ag(x)+b{)}^{-n})-\frac{y(x)f^{\prime }(x)}{f(x)}-f(x)g^{\prime }(x)+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 105.501 (sec), leaf count = 95

$$\text{Solve}[\frac{f(x)(ag(x)+b)\log (ag(x)+b){(f(x{)}^{-n}(ag(x)+b{)}^{-n})}^{\frac{1}{n}}}{a}+c_1={\int }_1^{y(x){(f(x{)}^{-n}(ag(x)+b{)}^{-n})}^{\frac{1}{n}}}\frac{1}{-{\left(a^n\right)}^{\frac{1}{n}}K[1]+K[1{]}^n+1}dK[1],y(x)]$$

✓ Maple : cpu = 0.07 (sec), leaf count = 281

$$\{y\left(x\right)=\frac{(ag\left(x\right)+b)f\left(x\right)}{a}\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-{\int }^{\mathit{\_}\mathit{Z}}\frac{{\left(\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right){(ag\left(x\right)+b)}^{-n}{\left(f\left(x\right)\right)}^{1-n}\right)}^{-n-1}{\left(f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^{-2n+1}{\left({\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^3{(ag\left(x\right)+b)}^{-n-1}a{\left(f\left(x\right)\right)}^{2-n}n\right)}^n{n}^{-n}}{\mathit{\_}\mathit{a}{\left(\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right){(ag\left(x\right)+b)}^{-n}{\left(f\left(x\right)\right)}^{1-n}\right)}^{-n-1}{\left(f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^{-2n+1}{\left({\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^3{(ag\left(x\right)+b)}^{-n-1}a{\left(f\left(x\right)\right)}^{2-n}n\right)}^n{n}^{-n}-{\left(\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right){(ag\left(x\right)+b)}^{-n}{\left(f\left(x\right)\right)}^{1-n}\right)}^{-n-1}{\left(f\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^{-2n+1}{\left({\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^3{(ag\left(x\right)+b)}^{-n-1}a{\left(f\left(x\right)\right)}^{2-n}n\right)}^n{n}^{-n}-{\mathit{\_}\mathit{a}}^n}d\mathit{\_}\mathit{a}-\ln (ag\left(x\right)+b)+\mathit{\_}\mathit{C}\mathit{1})\}$$