4.385   ODE No. 1385

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

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

Mathematica: cpu = 0.019002 (sec), leaf count = 78 $$\left\{\{y(x)\to c_1\sqrt{x^2+1}{P}_{\frac{1}{2}(\sqrt{1-a}-1)}^{\frac{1}{2}}(ix)+c_2\sqrt{x^2+1}{Q}_{\frac{1}{2}(\sqrt{1-a}-1)}^{\frac{1}{2}}(ix)\}\right\}$$

Maple: cpu = 0.047 (sec), leaf count = 61 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}\sqrt[4]{x^2+1}{(x+\sqrt{x^2+1})}^{\frac{1}{2}\sqrt{1-a}}+\mathit{\_}\mathit{C}\mathit{2}\sqrt[4]{x^2+1}{(x+\sqrt{x^2+1})}^{-\frac{1}{2}\sqrt{1-a}}\}$$