2.1561   ODE No. 1561

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

$$y(x)(a{x}^4+(n-2)n(n+1)(n+3))-2n(n+1)x^2{y}^{″}(x)+4n(n+1)x{y}^{\prime }(x)+x^4{y}^{(4)}(x)=0$$ ✓ Mathematica : cpu = 2.91633 (sec), leaf count = 400

$$\left\{\{y(x)\to c_1(-2^{n-\frac{5}{2}})\sqrt{x}{a}^{\frac{2-n}{4}+\frac{1}{4}(n-\frac{3}{2})}\mathrm{\Gamma }(\frac{3}{2}-n)(\cos \left(\frac{3}{4}\pi (\frac{3}{2}-n)\right){\text{ber}}_{-n-\frac{1}{2}}\left(\sqrt[4]{a}x\right)+\sin \left(\frac{3}{4}\pi (\frac{3}{2}-n)\right){\text{bei}}_{-n-\frac{1}{2}}\left(\sqrt[4]{a}x\right))+c_2{2}^{n-\frac{1}{2}}\sqrt{x}{a}^{\frac{1}{4}(n+\frac{1}{2})-\frac{n}{4}}\mathrm{\Gamma }(\frac{1}{2}-n)(\cos \left(\frac{3}{4}\pi (-n-\frac{1}{2})\right){\text{ber}}_{-n-\frac{1}{2}}\left(\sqrt[4]{a}x\right)+\sin \left(\frac{3}{4}\pi (-n-\frac{1}{2})\right){\text{bei}}_{-n-\frac{1}{2}}\left(\sqrt[4]{a}x\right))+c_3{2}^{-n-\frac{3}{2}}\sqrt{x}{a}^{\frac{1}{4}(-n-\frac{1}{2})+\frac{n+1}{4}}\mathrm{\Gamma }(n+\frac{3}{2})(\cos \left(\frac{3}{4}\pi (n+\frac{1}{2})\right){\text{ber}}_{n+\frac{1}{2}}\left(\sqrt[4]{a}x\right)+\sin \left(\frac{3}{4}\pi (n+\frac{1}{2})\right){\text{bei}}_{n+\frac{1}{2}}\left(\sqrt[4]{a}x\right))-c_4{2}^{-n-\frac{7}{2}}\sqrt{x}{a}^{\frac{1}{4}(-n-\frac{5}{2})+\frac{n+3}{4}}\mathrm{\Gamma }(n+\frac{5}{2})(\cos \left(\frac{3}{4}\pi (n+\frac{5}{2})\right){\text{ber}}_{n+\frac{1}{2}}\left(\sqrt[4]{a}x\right)+\sin \left(\frac{3}{4}\pi (n+\frac{5}{2})\right){\text{bei}}_{n+\frac{1}{2}}\left(\sqrt[4]{a}x\right))\}\right\}$$ ✓ Maple : cpu = 0.212 (sec), leaf count = 69

$$\{y\left(x\right)=\sqrt{x}({\mathit{J}}_{n+\frac{1}{2}}\left(\sqrt{-\sqrt{-a}}x\right)\mathit{\_}\mathit{C}\mathit{3}+{\mathit{Y}}_{n+\frac{1}{2}}\left(\sqrt[4]{-a}x\right)\mathit{\_}\mathit{C}\mathit{2}+{\mathit{J}}_{n+\frac{1}{2}}\left(\sqrt[4]{-a}x\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{Y}}_{n+\frac{1}{2}}\left(\sqrt{-\sqrt{-a}}x\right)\mathit{\_}\mathit{C}\mathit{4})\}$$