4.83   ODE No. 1083

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)}{f\left(x\right)}+(3/4\frac{{\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)}^2}{{\left(f\left(x\right)\right)}^2}-1/2\frac{\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}f\left(x\right)}{f\left(x\right)}-3/4\frac{{\left(\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}g\left(x\right)\right)}^2}{{\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^2}+1/2\frac{\frac{{\mathrm{d}}^3}{\mathrm{d}{x}^3}g\left(x\right)}{\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)}+\frac{(1/4-v^2){\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^2}{{\left(g\left(x\right)\right)}^2}+{\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)}^2)y\left(x\right)=0}}$$

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

Mathematica: cpu = 0.823605 (sec), leaf count = 111 $$\text{DSolve}[-\frac{f^{\prime }(x)y^{\prime }(x)}{f(x)}+y(x)(-\frac{f^{″}(x)}{2f(x)}+\frac{3f^{\prime }(x{)}^2}{4f(x{)}^2}+\frac{(\frac{1}{4}-v^2)g^{\prime }(x{)}^2}{g(x{)}^2}+g^{\prime }(x{)}^2+\frac{g^3(x)}{2g^{\prime }(x)}-\frac{3g^{″}(x{)}^2}{4g^{\prime }(x{)}^2})+y^{″}(x)=0,y(x),x]$$

Maple: cpu = 0.078 (sec), leaf count = 43 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}\sqrt{\frac{f\left(x\right)g\left(x\right)}{\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)}}{\mathit{J}}_v\left(g\left(x\right)\right)+\mathit{\_}\mathit{C}\mathit{2}\sqrt{\frac{f\left(x\right)g\left(x\right)}{\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)}}{\mathit{Y}}_v\left(g\left(x\right)\right)\}$$