2.144   ODE No. 144

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

$$x^2(ay(x{)}^2+y^{\prime }(x))+b{x}^{\alpha }+c=0$$ ✓ Mathematica : cpu = 0.217838 (sec), leaf count = 1787

$$\left\{\{y(x)\to \frac{a^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\alpha }^{-\frac{i\sqrt{4ac-1}\alpha +\alpha }{{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}+1}{b}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}(\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}){\left(x^{\alpha }\right)}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-1}{J}_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)\mathrm{\Gamma }(\frac{\sqrt{1-4ac}}{\alpha }+1)x^{\alpha -1}+\frac{1}{2}{a}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\alpha }^{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-\frac{i\sqrt{4ac-1}\alpha +\alpha }{{\alpha }^2}}{b}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\left(x^{\alpha }\right)}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-\frac{1}{2}}(J_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-1}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)-J_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}+1}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right))\mathrm{\Gamma }(\frac{\sqrt{1-4ac}}{\alpha }+1)x^{\alpha -1}+c_1(a^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\alpha }^{-\frac{\alpha -i\alpha \sqrt{4ac-1}}{{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}+1}{b}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}(\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}){\left(x^{\alpha }\right)}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-1}{J}_{-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)\mathrm{\Gamma }(1-\frac{\sqrt{1-4ac}}{\alpha })x^{\alpha -1}+\frac{1}{2}{a}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\alpha }^{-\frac{\alpha -i\alpha \sqrt{4ac-1}}{{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}{b}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\left(x^{\alpha }\right)}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-\frac{1}{2}}(J_{-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-1}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)-J_{1-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right))\mathrm{\Gamma }(1-\frac{\sqrt{1-4ac}}{\alpha })x^{\alpha -1})}{a(a^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{b}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\left(x^{\alpha }\right)}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{J}_{-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)c_1\mathrm{\Gamma }(1-\frac{\sqrt{1-4ac}}{\alpha }){\alpha }^{-\frac{\alpha -i\alpha \sqrt{4ac-1}}{{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}+a^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{b}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\left(x^{\alpha }\right)}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{J}_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)\mathrm{\Gamma }(\frac{\sqrt{1-4ac}}{\alpha }+1){\alpha }^{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-\frac{i\sqrt{4ac-1}\alpha +\alpha }{{\alpha }^2}})}\}\right\}$$ ✓ Maple : cpu = 0.095 (sec), leaf count = 219

$$\{y\left(x\right)=\frac{1}{2ax}(-2({\mathit{Y}}_{\frac{\sqrt{-4ac+1}+\alpha }{\alpha }}\left(2\frac{\sqrt{ab}{x}^{\alpha /2}}{\alpha }\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{J}}_{\frac{\sqrt{-4ac+1}+\alpha }{\alpha }}\left(2\frac{\sqrt{ab}{x}^{\alpha /2}}{\alpha }\right))\sqrt{ab}{x}^{\alpha /2}+(\sqrt{-4ac+1}+1)({\mathit{Y}}_{\frac{1}{\alpha }\sqrt{-4ac+1}}\left(2\frac{\sqrt{ab}{x}^{\alpha /2}}{\alpha }\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{J}}_{\frac{1}{\alpha }\sqrt{-4ac+1}}\left(2\frac{\sqrt{ab}{x}^{\alpha /2}}{\alpha }\right))){({\mathit{Y}}_{\frac{1}{\alpha }\sqrt{-4ac+1}}\left(2\frac{\sqrt{ab}{x}^{\alpha /2}}{\alpha }\right)\mathit{\_}\mathit{C}\mathit{1}+{\mathit{J}}_{\frac{1}{\alpha }\sqrt{-4ac+1}}\left(2\frac{\sqrt{ab}{x}^{\alpha /2}}{\alpha }\right))}^{-1}\}$$