4.220   ODE No. 1220

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

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

Mathematica: cpu = 11.805499 (sec), leaf count = 96 $$\left\{\{y(x)\to c_1{J}_{\frac{1}{2}(2v-1)}\left(\sqrt{a}x\right)\exp ({\int }_1^x\frac{1-2K[1]f(K[1])}{2K[1]}dK[1])+c_2{Y}_{\frac{1}{2}(2v-1)}\left(\sqrt{a}x\right)\exp ({\int }_1^x\frac{1-2K[1]f(K[1])}{2K[1]}dK[1])\}\right\}$$

Maple: cpu = 0.015 (sec), leaf count = 51 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{-\frac{\int 2f\left(x\right)\mathrm{d}x}{2}}\sqrt{x}{\mathit{J}}_{v-\frac{1}{2}}\left(\sqrt{a}x\right)+\mathit{\_}\mathit{C}\mathit{2}{\mathrm{e}}^{-\frac{\int 2f\left(x\right)\mathrm{d}x}{2}}\sqrt{x}{\mathit{Y}}_{v-\frac{1}{2}}\left(\sqrt{a}x\right)\}$$