3.394   ODE No. 394

$$\boxed{{\displaystyle {\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2+2f\left(x\right)y\left(x\right)\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+g\left(x\right){\left(y\left(x\right)\right)}^2-(g\left(x\right)-{\left(f\left(x\right)\right)}^2){\mathrm{e}}^{-2{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}=0}}$$

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

Mathematica: cpu = 58.132882 (sec), leaf count = 52 $$\text{DSolve}[-(g(x)-f(x{)}^2)e^{-2{\int }_a^{x}f(\text{xp})d\text{xp}}+2f(x)y(x)y^{\prime }(x)+g(x)y(x{)}^2+y^{\prime }(x{)}^2=0,y(x),x]$$

Maple: cpu = 3.822 (sec), leaf count = 310 $$\{y\left(x\right)=-\tan \left(\frac{1}{2\cos \left(2\right)+2}(-2\mathit{\_}\mathit{C}\mathit{1}\cos \left(2\right)+\sqrt{2}\int {\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^2\sqrt{-\frac{\cos \left(4\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}-4\frac{\cos \left(2\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}+\frac{\cos \left(4\right)g\left(x\right)}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}+4\frac{\cos \left(2\right)g\left(x\right)}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}-3\frac{{\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}+3\frac{g\left(x\right)}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}}\mathrm{d}x-2\mathit{\_}\mathit{C}\mathit{1})\right)\sqrt{{\mathrm{e}}^{-2{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}{({(\tan \left(\frac{1}{2\cos \left(2\right)+2}(-2\mathit{\_}\mathit{C}\mathit{1}\cos \left(2\right)+\sqrt{2}\int {\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^2\sqrt{-\frac{\cos \left(4\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}-4\frac{\cos \left(2\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}+\frac{\cos \left(4\right)g\left(x\right)}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}+4\frac{\cos \left(2\right)g\left(x\right)}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}-3\frac{{\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}+3\frac{g\left(x\right)}{{\left({\mathrm{e}}^{{\int }_a^{x}f\left(\mathit{x}\mathit{p}\right)\mathrm{d}\mathit{x}\mathit{p}}\right)}^4}}\mathrm{d}x-2\mathit{\_}\mathit{C}\mathit{1})\right))}^2+1)}^{-1}}\}$$