4.55   ODE No. 1055

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)+(ax+b)\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+(\mathit{a}\mathit{1}{x}^2+\mathit{b}\mathit{1}x+\mathit{c}\mathit{1})y\left(x\right)=0}}$$

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

Mathematica: cpu = 0.155020 (sec), leaf count = 421 $$\left\{\{y(x)\to c_1\exp \left(\frac{-bx\sqrt{a^2-4\text{a1}}-\frac{1}{2}a{x}^2\sqrt{a^2-4\text{a1}}-\frac{1}{2}{a}^2{x}^2-abx+2\text{a1}{x}^2+2\text{b1}x}{2\sqrt{a^2-4\text{a1}}}\right)H_{\frac{-a^3-a^2\sqrt{a^2-4\text{a1}}+4\text{a1}\sqrt{a^2-4\text{a1}}+2a^2\text{c1}+4a\text{a1}-2ab\text{b1}+2\text{a1}{b}^2-8\text{a1}\text{c1}+2{\text{b1}}^2}{2{(a^2-4\text{a1})}^{3/2}}}(\frac{ab-2\text{b1}}{\sqrt{2}{(a^2-4\text{a1})}^{3/4}}+\frac{x\sqrt[4]{a^2-4\text{a1}}}{\sqrt{2}})+c_2\exp \left(\frac{-bx\sqrt{a^2-4\text{a1}}-\frac{1}{2}a{x}^2\sqrt{a^2-4\text{a1}}-\frac{1}{2}{a}^2{x}^2-abx+2\text{a1}{x}^2+2\text{b1}x}{2\sqrt{a^2-4\text{a1}}}\right){}_1{F}_1(-\frac{-a^3+2\text{c1}{a}^2-\sqrt{a^2-4\text{a1}}{a}^2+4\text{a1}a-2b\text{b1}a+2\text{a1}{b}^2+2{\text{b1}}^2+4\sqrt{a^2-4\text{a1}}\text{a1}-8\text{a1}\text{c1}}{4{(a^2-4\text{a1})}^{3/2}};\frac{1}{2};{(\frac{ab-2\text{b1}}{\sqrt{2}{(a^2-4\text{a1})}^{3/4}}+\frac{\sqrt[4]{a^2-4\text{a1}}x}{\sqrt{2}})}^2)\}\right\}$$

Maple: cpu = 0.172 (sec), leaf count = 317 $$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{}_1\text{F}{}_1(\frac{1}{4}({(a^2-4\mathit{a}\mathit{1})}^{\frac{3}{2}}+a^3-2a^2\mathit{c}\mathit{1}+(2\mathit{b}\mathit{1}b-4\mathit{a}\mathit{1})a+(-2b^2+8\mathit{c}\mathit{1})\mathit{a}\mathit{1}-2{\mathit{b}\mathit{1}}^2){(a^2-4\mathit{a}\mathit{1})}^{-\frac{3}{2}};\frac{1}{2};\frac{{(a^{2}x+ab-4\mathit{a}\mathit{1}x-2\mathit{b}\mathit{1})}^2}{2}{(a^2-4\mathit{a}\mathit{1})}^{-\frac{3}{2}}){\mathrm{e}}^{-\frac{x}{4}((ax+2b){(a^2-4\mathit{a}\mathit{1})}^{\frac{3}{2}}+(a^2-4\mathit{a}\mathit{1})(a^{2}x+2ab-4\mathit{a}\mathit{1}x-4\mathit{b}\mathit{1})){(a^2-4\mathit{a}\mathit{1})}^{-\frac{3}{2}}}+\mathit{\_}\mathit{C}\mathit{2}{}_1\text{F}{}_1(\frac{1}{4}(3{(a^2-4\mathit{a}\mathit{1})}^{3/2}+a^3-2a^2\mathit{c}\mathit{1}+(2\mathit{b}\mathit{1}b-4\mathit{a}\mathit{1})a+(-2b^2+8\mathit{c}\mathit{1})\mathit{a}\mathit{1}-2{\mathit{b}\mathit{1}}^2){(a^2-4\mathit{a}\mathit{1})}^{-\frac{3}{2}};\frac{3}{2};\frac{{(a^{2}x+ab-4\mathit{a}\mathit{1}x-2\mathit{b}\mathit{1})}^2}{2}{(a^2-4\mathit{a}\mathit{1})}^{-\frac{3}{2}})(a^{2}x+ab-4\mathit{a}\mathit{1}x-2\mathit{b}\mathit{1}){\mathrm{e}}^{-\frac{x}{4}((ax+2b){(a^2-4\mathit{a}\mathit{1})}^{\frac{3}{2}}+(a^2-4\mathit{a}\mathit{1})(a^{2}x+2ab-4\mathit{a}\mathit{1}x-4\mathit{b}\mathit{1})){(a^2-4\mathit{a}\mathit{1})}^{-\frac{3}{2}}}\}$$