6.36   ODE No. 1569

$$\boxed{{\displaystyle x^4\mathit{d}\mathit{4}\mathit{y}\left(x\right)+(6-4a)x^3\frac{{\mathrm{d}}^3}{\mathrm{d}{x}^3}y\left(x\right)+(4b^2{c}^2{x}^{2c}+6{(a-1)}^2-2c^2({\mu }^2+{\nu }^2)+1)x^2\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)+(4(3c-2a+1)b^2{c}^2{x}^{2c}+(2a-1)(2c^2({\mu }^2+{\nu }^2)-2a(a-1)-1))x\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+(4(a-c)(a-2c)b^2{c}^2{x}^{2c}+(c\mu +c\nu +a)(c\mu +c\nu -a)(c\mu -c\nu +a)(c\mu -c\nu -a))y\left(x\right)=0}}$$

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

Mathematica: cpu = 0.152019 (sec), leaf count = 378 $$\left\{\{y(x)\to c_1{b}^{\frac{a-c\mu -c\nu }{c}}{\left(x^{2c}\right)}^{\frac{a-c\mu -c\nu }{2c}}{}_2{F}_3(-\frac{\mu }{2}-\frac{\nu }{2}+\frac{1}{2},-\frac{\mu }{2}-\frac{\nu }{2}+1;1-\mu ,1-\nu ,-\mu -\nu +1;-b^2{x}^{2c})+c_2{b}^{\frac{a+c\mu -c\nu }{c}}{\left(x^{2c}\right)}^{\frac{a+c\mu -c\nu }{2c}}{}_2{F}_3(\frac{\mu }{2}-\frac{\nu }{2}+\frac{1}{2},\frac{\mu }{2}-\frac{\nu }{2}+1;\mu +1,1-\nu ,\mu -\nu +1;-b^2{x}^{2c})+c_3{b}^{\frac{a-c\mu +c\nu }{c}}{\left(x^{2c}\right)}^{\frac{a-c\mu +c\nu }{2c}}{}_2{F}_3(-\frac{\mu }{2}+\frac{\nu }{2}+\frac{1}{2},-\frac{\mu }{2}+\frac{\nu }{2}+1;1-\mu ,\nu +1,-\mu +\nu +1;-b^2{x}^{2c})+c_4{b}^{\frac{a+c\mu +c\nu }{c}}{\left(x^{2c}\right)}^{\frac{a+c\mu +c\nu }{2c}}{}_2{F}_3(\frac{\mu }{2}+\frac{\nu }{2}+\frac{1}{2},\frac{\mu }{2}+\frac{\nu }{2}+1;\mu +1,\nu +1,\mu +\nu +1;-b^2{x}^{2c})\}\right\}$$

Maple: cpu = 0.218 (sec), leaf count = 81 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{x}^a{\mathit{J}}_{\mu }\left(x^{c}b\right){\mathit{Y}}_{\nu }\left(x^{c}b\right)+\mathit{\_}\mathit{C}\mathit{2}{x}^a{\mathit{J}}_{\nu }\left(x^{c}b\right){\mathit{J}}_{\mu }\left(x^{c}b\right)+\mathit{\_}\mathit{C}\mathit{3}{x}^a{\mathit{J}}_{\nu }\left(x^{c}b\right){\mathit{Y}}_{\mu }\left(x^{c}b\right)+\mathit{\_}\mathit{C}\mathit{4}{x}^a{\mathit{Y}}_{\nu }\left(x^{c}b\right){\mathit{Y}}_{\mu }\left(x^{c}b\right)\}$$