2.1248   ODE No. 1248

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

$$ax{y}^{\prime }(x)+y(x)(b{x}^2+cx+d)+(x^2-1)y^{″}(x)=0$$ ✓ Mathematica : cpu = 0.251891 (sec), leaf count = 238

$$\left\{\{y(x)\to c_2{(\frac{x}{2}-\frac{1}{2})}^{a/4}{(x^2-1)}^{-a/4}{(\frac{x}{2}+\frac{1}{2})}^{1-\frac{a}{4}}{e}^{\sqrt{-b}x}\text{HeunC}[\frac{1}{4}a(a-4\sqrt{-b}-2)-b+4\sqrt{-b}+c-d,2(2\sqrt{-b}+c),2-\frac{a}{2},\frac{a}{2},4\sqrt{-b},\frac{x}{2}+\frac{1}{2}]+c_1((x-1)(x+1){)}^{a/4}{(x^2-1)}^{-a/4}{e}^{\sqrt{-b}x}\text{HeunC}[a\sqrt{-b}-b+c-d,2(a\sqrt{-b}+c),\frac{a}{2},\frac{a}{2},4\sqrt{-b},\frac{x}{2}+\frac{1}{2}]\}\right\}$$ ✓ Maple : cpu = 1.224 (sec), leaf count = 134

$$\{y\left(x\right)={\mathrm{e}}^{\sqrt{-b}x}{(x^2-1)}^{-\frac{a}{4}}({(\frac{1}{2}+\frac{x}{2})}^{1-\frac{a}{4}}{(-\frac{1}{2}+\frac{x}{2})}^{\frac{a}{4}}\mathit{H}\mathit{e}\mathit{u}\mathit{n}\mathit{C}(4\sqrt{-b},1-\frac{a}{2},\frac{a}{2}-1,2c,d-c-\frac{a^2}{8}+b+\frac{1}{2},\frac{1}{2}+\frac{x}{2})\mathit{\_}\mathit{C}\mathit{2}+\mathit{H}\mathit{e}\mathit{u}\mathit{n}\mathit{C}(4\sqrt{-b},\frac{a}{2}-1,\frac{a}{2}-1,2c,d-c-\frac{a^2}{8}+b+\frac{1}{2},\frac{1}{2}+\frac{x}{2}){\left((1+x)(x-1)\right)}^{\frac{a}{4}}\mathit{\_}\mathit{C}\mathit{1})\}$$