4.389   ODE No. 1389

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

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

Mathematica: cpu = 0.378548 (sec), leaf count = 217 $$\left\{\{y(x)\to c_2(-1{)}^{\frac{1}{2}(-2v-3)+1}{x}^{\frac{1}{4}(-2v-3)+1}{e}^{\frac{1}{4}(-2\log (1-x)-\log (x))}(x-1{)}^{\frac{1}{2}(n+\frac{1}{2}(2n+1)+\frac{1}{2}(-2v-3)+v+2)}{}_2{F}_1(\frac{1}{2}(2n+1)+\frac{1}{2}(-2v-3)+1,n+\frac{1}{2}(-2v-3)+v+2;\frac{1}{2}(-2v-3)+2;x)+c_1{x}^{\frac{1}{4}(2v+3)}{e}^{\frac{1}{4}(-2\log (1-x)-\log (x))}(x-1{)}^{\frac{1}{2}(n+\frac{1}{2}(2n+1)+\frac{1}{2}(-2v-3)+v+2)}{}_2{F}_1(\frac{1}{2}(2n+1),n+v+1;\frac{1}{2}(2v+3);x)\}\right\}$$

Maple: cpu = 0.047 (sec), leaf count = 74 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{x}^{-\frac{v}{2}}{(x-1)}^{-n}{}_2\text{F}{}_1(-v-n,-n+\frac{1}{2};\frac{1}{2}-v;x)+\mathit{\_}\mathit{C}\mathit{2}{x}^{\frac{1}{2}+\frac{v}{2}}{(x-1)}^{-n}{}_2\text{F}{}_1(v-n+1,-n+\frac{1}{2};\frac{3}{2}+v;x)\}$$