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 = 3.9508 (sec), leaf count = 516

$$\left\{\{y(x)\to -\frac{x^{\nu }(\sqrt{b}{c}_1(\nu +1)\sqrt{(\nu +1{)}^2}U(\frac{1}{2}(\frac{\sqrt{a}c}{\sqrt{b}\sqrt{(\nu +1{)}^2}}+\frac{\nu }{\nu +1}),\frac{\nu }{\nu +1},\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{(\nu +1{)}^2}})+c_1(\sqrt{a}c(\nu +1)+\sqrt{b}\sqrt{(\nu +1{)}^2}\nu )U(\frac{1}{2}(\frac{\sqrt{a}c}{\sqrt{b}\sqrt{(\nu +1{)}^2}}+\frac{3\nu +2}{\nu +1}),\frac{\nu }{\nu +1}+1,\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{(\nu +1{)}^2}})+\sqrt{b}(\nu +1)\sqrt{(\nu +1{)}^2}(L_{-\frac{\sqrt{a}c}{2\sqrt{b}\sqrt{(\nu +1{)}^2}}-\frac{\nu }{2(\nu +1)}}^{-\frac{1}{\nu +1}}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{(\nu +1{)}^2}}\right)+2L_{-\frac{\sqrt{a}c}{2\sqrt{b}\sqrt{(\nu +1{)}^2}}-\frac{3\nu +2}{2\nu +2}}^{\frac{\nu }{\nu +1}}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{(\nu +1{)}^2}}\right)))}{\sqrt{a}(\nu +1{)}^2(c_{1}U(\frac{1}{2}(\frac{\sqrt{a}c}{\sqrt{b}\sqrt{(\nu +1{)}^2}}+\frac{\nu }{\nu +1}),\frac{\nu }{\nu +1},\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{(\nu +1{)}^2}})+L_{-\frac{\sqrt{a}c}{2\sqrt{b}\sqrt{(\nu +1{)}^2}}-\frac{\nu }{2(\nu +1)}}^{-\frac{1}{\nu +1}}\left(\frac{2\sqrt{a}\sqrt{b}{x}^{\nu +1}}{\sqrt{(\nu +1{)}^2}}\right))}\}\right\}$$

✓ Maple : cpu = 0.473 (sec), leaf count = 348

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

Hand solution

$$\begin{array}{rl}{y}^{\prime }+a{y}^2-b{x}^{2v}-c{x}^{v-1}& =0\\ \text{(1)}& y^{\prime }& =b{x}^v+c{x}^{v-1}-a{y}^2& =P\left(x\right)+Q\left(x\right)y+R\left(x\right)y^2\end{array}$$

This is Riccati first order non-linear ODE with $P\left(x\right)=b{x}^v+c{x}^{v-1},Q\left(x\right)=0,R\left(x\right)=-a$.

Need to do this later.