2.25   ODE No. 25

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

$$ay(x{)}^2-b{x}^{2\nu }-c{x}^{\nu -1}+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.232857 (sec), leaf count = 1835

$$\left\{\{y(x)\to \frac{-2^{\frac{\nu }{2(\nu +1)}-1}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}\nu {\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}{L}_{-\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)}}^{\frac{\nu }{\nu +1}-1}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}\right)x^{-\frac{\nu }{2}-1}-\frac{2^{\frac{\nu }{2(\nu +1)}}\sqrt{a}\sqrt{b}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}(\nu +1){\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}{L}_{-\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)}}^{\frac{\nu }{\nu +1}-1}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}\right)x^{\nu /2}}{\sqrt{{\nu }^2+2\nu +1}}+2^{\frac{\nu }{2(\nu +1)}-1}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}\nu {\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}-1}{L}_{-\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)}}^{\frac{\nu }{\nu +1}-1}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}\right)x^{\nu /2}-\frac{2^{\frac{\nu }{2(\nu +1)}+1}\sqrt{a}\sqrt{b}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}(\nu +1){\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}{L}_{-\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)}-1}^{\frac{\nu }{\nu +1}}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}\right)x^{\nu /2}}{\sqrt{{\nu }^2+2\nu +1}}+c_1(-2^{\frac{\nu }{2(\nu +1)}-1}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}\nu {\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}U(\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)},\frac{\nu }{\nu +1},\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}})x^{-\frac{\nu }{2}-1}-\frac{2^{\frac{\nu }{2(\nu +1)}}\sqrt{a}\sqrt{b}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}(\nu +1){\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}U(\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)},\frac{\nu }{\nu +1},\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}})x^{\nu /2}}{\sqrt{{\nu }^2+2\nu +1}}+2^{\frac{\nu }{2(\nu +1)}-1}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}\nu {\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}-1}U(\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)},\frac{\nu }{\nu +1},\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}})x^{\nu /2}-\frac{2^{\frac{\nu }{2(\nu +1)}}\sqrt{a}\sqrt{b}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}(\nu +1)(\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu ){\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}U(\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)}+1,\frac{\nu }{\nu +1}+1,\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}})x^{\nu /2}}{(\nu b+b)\sqrt{{\nu }^2+2\nu +1}})}{a(2^{\frac{\nu }{2(\nu +1)}}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}{\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}{c}_{1}U(\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)},\frac{\nu }{\nu +1},\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}})x^{-\nu /2}+2^{\frac{\nu }{2(\nu +1)}}{e}^{-\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}}{\left(x^{\nu +1}\right)}^{\frac{\nu }{2(\nu +1)}}{L}_{-\frac{\frac{\sqrt{a}\sqrt{b}\nu c}{\sqrt{(\nu +1{)}^2}}+\frac{\sqrt{a}\sqrt{b}c}{\sqrt{(\nu +1{)}^2}}+b\nu }{2(\nu b+b)}}^{\frac{\nu }{\nu +1}-1}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{{\nu }^2+2\nu +1}}\right)x^{-\nu /2})}\}\right\}$$

✓ Maple : cpu = 0.307 (sec), leaf count = 378

$$\{y\left(x\right)=\frac{1}{2ax}((2\sqrt{a}{x}^{\nu +1}\mathit{\_}\mathit{C}\mathit{1}{b}^2-b^{\frac{3}{2}}\mathit{\_}\mathit{C}\mathit{1}\nu +\sqrt{a}\mathit{\_}\mathit{C}\mathit{1}bc){\mathit{W}}_{-\frac{c}{2\nu +2}\sqrt{a}\frac{1}{\sqrt{b}},\frac{1}{2\nu +2}}\left(2\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\nu +1}\right)+(-2b^{3/2}\mathit{\_}\mathit{C}\mathit{1}\nu -2b^{3/2}\mathit{\_}\mathit{C}\mathit{1}){\mathit{W}}_{-\frac{1}{2\nu +2}(\sqrt{a}c-2\sqrt{b}\nu -2\sqrt{b})\frac{1}{\sqrt{b}},\frac{1}{2\nu +2}}\left(2\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\nu +1}\right)+(2\sqrt{a}{x}^{\nu +1}{b}^2-b^{\frac{3}{2}}\nu +\sqrt{a}bc){\mathit{M}}_{-\frac{c}{2\nu +2}\sqrt{a}\frac{1}{\sqrt{b}},\frac{1}{2\nu +2}}\left(2\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\nu +1}\right)+(b^{\frac{3}{2}}\nu -\sqrt{a}bc+2b^{3/2}){\mathit{M}}_{-\frac{1}{2\nu +2}(\sqrt{a}c-2\sqrt{b}\nu -2\sqrt{b})\frac{1}{\sqrt{b}},\frac{1}{2\nu +2}}\left(2\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\nu +1}\right))b^{-\frac{3}{2}}{({\mathit{W}}_{-\frac{c}{2\nu +2}\sqrt{a}\frac{1}{\sqrt{b}},\frac{1}{2\nu +2}}\left(2\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\nu +1}\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{M}}_{-\frac{c}{2\nu +2}\sqrt{a}\frac{1}{\sqrt{b}},\frac{1}{2\nu +2}}\left(2\frac{\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\nu +1}\right))}^{-1}\}$$