6.32   ODE No. 1565

$$\boxed{{\displaystyle x^4\mathit{d}\mathit{4}\mathit{y}\left(x\right)+6x^3\frac{{\mathrm{d}}^3}{\mathrm{d}{x}^3}y\left(x\right)+(4x^4+(-{\rho }^2-{\sigma }^2+7)x^2)\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)+(16x^3+(-{\rho }^2-{\sigma }^2+1)x)\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+({\rho }^2{\sigma }^2+8x^2)y\left(x\right)=0}}$$

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

Mathematica: cpu = 0.518066 (sec), leaf count = 242 $$\left\{\{y(x)\to c_1{x}^{-\rho }{}_2{F}_3(\frac{1}{2}-\frac{\rho }{2},1-\frac{\rho }{2};1-\rho ,-\frac{\rho }{2}-\frac{\sigma }{2}+1,-\frac{\rho }{2}+\frac{\sigma }{2}+1;-x^2)+c_3{x}^{-\sigma }{}_2{F}_3(\frac{1}{2}-\frac{\sigma }{2},1-\frac{\sigma }{2};1-\sigma ,-\frac{\rho }{2}-\frac{\sigma }{2}+1,\frac{\rho }{2}-\frac{\sigma }{2}+1;-x^2)+c_4{x}^{\sigma }{}_2{F}_3(\frac{\sigma }{2}+\frac{1}{2},\frac{\sigma }{2}+1;-\frac{\rho }{2}+\frac{\sigma }{2}+1,\frac{\rho }{2}+\frac{\sigma }{2}+1,\sigma +1;-x^2)+c_2{x}^{\rho }{}_2{F}_3(\frac{\rho }{2}+\frac{1}{2},\frac{\rho }{2}+1;\rho +1,\frac{\rho }{2}-\frac{\sigma }{2}+1,\frac{\rho }{2}+\frac{\sigma }{2}+1;-x^2)\}\right\}$$

Maple: cpu = 0.171 (sec), leaf count = 85 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{\mathit{J}}_{\frac{\rho }{2}+\frac{\sigma }{2}}\left(x\right){\mathit{J}}_{\frac{\rho }{2}-\frac{\sigma }{2}}\left(x\right)+\mathit{\_}\mathit{C}\mathit{2}{\mathit{J}}_{\frac{\rho }{2}+\frac{\sigma }{2}}\left(x\right){\mathit{Y}}_{\frac{\rho }{2}-\frac{\sigma }{2}}\left(x\right)+\mathit{\_}\mathit{C}\mathit{3}{\mathit{Y}}_{\frac{\rho }{2}+\frac{\sigma }{2}}\left(x\right){\mathit{J}}_{\frac{\rho }{2}-\frac{\sigma }{2}}\left(x\right)+\mathit{\_}\mathit{C}\mathit{4}{\mathit{Y}}_{\frac{\rho }{2}+\frac{\sigma }{2}}\left(x\right){\mathit{Y}}_{\frac{\rho }{2}-\frac{\sigma }{2}}\left(x\right)\}$$