2.1215   ODE No. 1215

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

$$x{y}^{\prime }(x)(a{x}^n+b)+y(x)(\text{a1}{x}^{2n}+\text{b1}{x}^n+\text{c1})+x^2{y}^{″}(x)=0$$ ✓ Mathematica : cpu = 0.161466 (sec), leaf count = 664

$$\left\{\{y(x)\to c_1{x}^{\frac{1-n}{2}}{2}^{\frac{\sqrt{b^2{n}^2-2b{n}^2-4\text{c1}{n}^2+n^2}+n^2}{2n^2}}{\left(x^n\right)}^{\frac{\sqrt{b^2{n}^2-2b{n}^2-4\text{c1}{n}^2+n^2}+n^2}{2n^2}}\exp (\frac{1}{2}(-\frac{a{x}^n}{n}-b\log (x))-\frac{\sqrt{a^2-4\text{a1}}{x}^n}{2n})U(\frac{\frac{\sqrt{(b^2-2b-4\text{c1}+1)n^2}{a}^2}{n^2}+a^2+\sqrt{a^2-4\text{a1}}a+\frac{\sqrt{a^2-4\text{a1}}ba}{n}-\frac{\sqrt{a^2-4\text{a1}}a}{n}-4\text{a1}-\frac{4\text{a1}\sqrt{(b^2-2b-4\text{c1}+1)n^2}}{n^2}-\frac{2\sqrt{a^2-4\text{a1}}\text{b1}}{n}}{2(a^2-4\text{a1})},\frac{n^2+\sqrt{b^2{n}^2-2b{n}^2-4\text{c1}{n}^2+n^2}}{n^2},\frac{\sqrt{a^2-4\text{a1}}{x}^n}{n})+c_2{x}^{\frac{1-n}{2}}{2}^{\frac{\sqrt{b^2{n}^2-2b{n}^2-4\text{c1}{n}^2+n^2}+n^2}{2n^2}}{\left(x^n\right)}^{\frac{\sqrt{b^2{n}^2-2b{n}^2-4\text{c1}{n}^2+n^2}+n^2}{2n^2}}\exp (\frac{1}{2}(-\frac{a{x}^n}{n}-b\log (x))-\frac{\sqrt{a^2-4\text{a1}}{x}^n}{2n})L_{-\frac{\frac{ab\sqrt{a^2-4\text{a1}}}{n}-\frac{2\text{b1}\sqrt{a^2-4\text{a1}}}{n}-\frac{a\sqrt{a^2-4\text{a1}}}{n}+a\sqrt{a^2-4\text{a1}}+\frac{a^2\sqrt{n^2(b^2-2b-4\text{c1}+1)}}{n^2}+a^2-\frac{4\text{a1}\sqrt{n^2(b^2-2b-4\text{c1}+1)}}{n^2}-4\text{a1}}{2(a^2-4\text{a1})}}^{\frac{\sqrt{b^2{n}^2-2b{n}^2-4\text{c1}{n}^2+n^2}+n^2}{n^2}-1}\left(\frac{\sqrt{a^2-4\text{a1}}{x}^n}{n}\right)\}\right\}$$

✓ Maple : cpu = 0.196 (sec), leaf count = 167

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