2.394   ODE No. 394

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

$$-(g(x)-f(x{)}^2)e^{-2{\int }_a^{x}f(\text{xp})d\text{xp}}+2f(x)y(x)y^{\prime }(x)+g(x)y(x{)}^2+y^{\prime }(x{)}^2=0$$ ✗ Mathematica : cpu = 88.2118 (sec), leaf count = 0 , could not solve

DSolve[-((-f[x]^2 + g[x])/E^(2*Integrate[f[xp], {xp, a, x}])) + g[x]*y[x]^2 + 2*f[x]*y[x]*Derivative[1][y][x] + Derivative[1][y][x]^2 == 0, y[x], x]

✓ Maple : cpu = 10.447 (sec), leaf count = 109

$$\{y\left(x\right)=\tan (-\int {\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^2\sqrt{\frac{g\left(x\right)-{\left(f\left(x\right)\right)}^2}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})\sqrt{{\mathrm{e}}^{-2{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}{({(\tan (-\int {\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^2\sqrt{\frac{g\left(x\right)-{\left(f\left(x\right)\right)}^2}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1}))}^2+1)}^{-1}}\}$$