8.58   ODE No. 1648

$$\boxed{{\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)-k{x}^a{\left(y\left(x\right)\right)}^b{\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^c=0}}$$

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

Mathematica: cpu = 0.086011 (sec), leaf count = 27 $$\text{DSolve}[y^{″}(x)-k{x}^{a}y(x{)}^b{y}^{\prime }(x{)}^c=0,y(x),x]$$

Maple: cpu = 1.404 (sec), leaf count = 413 $$\{y\left(x\right)=\mathit{O}\mathit{D}\mathit{E}\mathit{S}\mathit{o}\mathit{l}\mathit{S}\mathit{t}\mathit{r}\mathit{u}\mathit{c}(\mathit{\_}\mathit{a}{\mathrm{e}}^{\int \mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}},[\{\frac{\mathrm{d}}{\mathrm{d}\mathit{\_}\mathit{a}}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)=-\frac{{\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^3}{{(a-c+2)}^2}({\mathit{\_}\mathit{a}}^b{(-\frac{(a-c+2)(\mathit{\_}\mathit{a}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+1)}{\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)(b+c-1)})}^c{b}^{2}k+2{\mathit{\_}\mathit{a}}^b{(-\frac{(a-c+2)(\mathit{\_}\mathit{a}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+1)}{\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)(b+c-1)})}^{c}bck+{\mathit{\_}\mathit{a}}^b{(-\frac{(a-c+2)(\mathit{\_}\mathit{a}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+1)}{\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)(b+c-1)})}^c{c}^{2}k-2{\mathit{\_}\mathit{a}}^b{(-\frac{(a-c+2)(\mathit{\_}\mathit{a}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+1)}{\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)(b+c-1)})}^{c}bk-2{\mathit{\_}\mathit{a}}^b{(-\frac{(a-c+2)(\mathit{\_}\mathit{a}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+1)}{\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)(b+c-1)})}^{c}ck+{\mathit{\_}\mathit{a}}^b{(-\frac{(a-c+2)(\mathit{\_}\mathit{a}\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)+1)}{\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)(b+c-1)})}^{c}k-\mathit{\_}\mathit{a}{a}^2-\mathit{\_}\mathit{a}ab+ac\mathit{\_}\mathit{a}+\mathit{\_}\mathit{a}cb-3a\mathit{\_}\mathit{a}-2b\mathit{\_}\mathit{a}+c\mathit{\_}\mathit{a}-2\mathit{\_}\mathit{a})+\frac{(2a+b-c+3){\left(\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\right)}^2}{a-c+2}\},\{\mathit{\_}\mathit{a}=y\left(x\right)x^{\frac{a-c+2}{b+c-1}},\mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)=-\frac{a-c+2}{bx\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+x\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)c+ay\left(x\right)-cy\left(x\right)-x\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+2y\left(x\right)}{\left(x^{\frac{a-c+2}{b+c-1}}\right)}^{-1}\},\{x={\mathrm{e}}^{-\frac{(\int \mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1})(b+c-1)}{a-c+2}},y\left(x\right)=\mathit{\_}\mathit{a}{\mathrm{e}}^{\int \mathit{\_}\mathit{b}\left(\mathit{\_}\mathit{a}\right)\mathrm{d}\mathit{\_}\mathit{a}+\mathit{\_}\mathit{C}\mathit{1}}\}])\}$$