8.216   ODE No. 1806

$$\boxed{{\displaystyle -2xy\left(x\right)(1-x)(1-y\left(x\right))(x-y\left(x\right))\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}y\left(x\right)+x(1-x)(x-2xy\left(x\right)-2y\left(x\right)+3{\left(y\left(x\right)\right)}^2){\left(\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)\right)}^2+2y\left(x\right)(1-y\left(x\right))(x^2+y\left(x\right)-2xy\left(x\right))\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)-{\left(y\left(x\right)\right)}^2{(1-y\left(x\right))}^2-f{\left(y\left(x\right)(-1+y\left(x\right))(y\left(x\right)-x)\right)}^{3/2}=0}}$$

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

Mathematica: cpu = 18.424840 (sec), leaf count = 123 $$\text{DSolve}[-f(x)((y(x)-1)y(x)(y(x)-x){)}^{3/2}+2(1-y(x))(x^2-2xy(x)+y(x))y(x)y^{\prime }(x)-2(1-x)x(1-y(x))(x-y(x))y(x)y^{″}(x)+(1-x)x(3y(x{)}^2-2xy(x)-2y(x)+x)y^{\prime }(x{)}^2-(1-y(x){)}^{2}y(x{)}^2=0,y(x),x]$$

Maple: cpu = 3.073 (sec), leaf count = 916 $$\{(\frac{3}{4}\int \frac{1}{x-1}{\mathrm{e}}^{\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\int x^{\frac{3}{2}}{\int }^{y\left(x\right)}\frac{1}{{(-\mathit{\_}\mathit{a}+x)}^2}\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^3-{\mathit{\_}\mathit{a}}^{2}x-{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{a}x}}d\mathit{\_}\mathit{a}{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\mathrm{d}x\mathrm{d}x+\int \frac{1}{x-1}{\mathrm{e}}^{\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\int \sqrt{x}{\int }^{y\left(x\right)}\frac{1}{\mathit{\_}\mathit{a}(\mathit{\_}\mathit{a}-1){(-\mathit{\_}\mathit{a}+x)}^2}\sqrt{{\mathit{\_}\mathit{a}}^3-{\mathit{\_}\mathit{a}}^{2}x-{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{a}x}d\mathit{\_}\mathit{a}{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\mathrm{d}x\mathrm{d}x-\frac{3}{4}\int \frac{1}{x-1}{\mathrm{e}}^{\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\int \sqrt{x}{\int }^{y\left(x\right)}\frac{1}{{(-\mathit{\_}\mathit{a}+x)}^2}\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^3-{\mathit{\_}\mathit{a}}^{2}x-{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{a}x}}d\mathit{\_}\mathit{a}{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\mathrm{d}x\mathrm{d}x-\frac{1}{2}\int \frac{1}{x-1}{\mathrm{e}}^{\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\int 1{\int }^{y\left(x\right)}\frac{1}{\mathit{\_}\mathit{a}(\mathit{\_}\mathit{a}-1){(-\mathit{\_}\mathit{a}+x)}^2}\sqrt{{\mathit{\_}\mathit{a}}^3-{\mathit{\_}\mathit{a}}^{2}x-{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{a}x}d\mathit{\_}\mathit{a}{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\frac{1}{\sqrt{x}}\mathrm{d}x\mathrm{d}x+\frac{1}{4}\int \frac{1}{x-1}{\mathrm{e}}^{\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\int 1{\int }^{y\left(x\right)}\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^3-{\mathit{\_}\mathit{a}}^{2}x-{\mathit{\_}\mathit{a}}^2+\mathit{\_}\mathit{a}x}}d\mathit{\_}\mathit{a}{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\frac{1}{\sqrt{x}}\mathrm{d}x\mathrm{d}x+\frac{f}{2}\int \frac{1}{x-1}{\mathrm{e}}^{\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\int 1{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\frac{1}{\sqrt{x}}\mathrm{d}x\mathrm{d}x+\frac{{\left(y\left(x\right)\right)}^4}{2}{\int }^x\frac{1}{\mathit{\_}\mathit{f}-1}{\mathrm{e}}^{\int \frac{1}{\mathit{\_}\mathit{f}(\mathit{\_}\mathit{f}-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{\mathit{\_}\mathit{f}}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{\mathit{\_}\mathit{f}}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{f}}\int 1{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{\mathit{\_}\mathit{f}(\mathit{\_}\mathit{f}-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{\mathit{\_}\mathit{f}}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{\mathit{\_}\mathit{f}}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{f}}\frac{1}{\sqrt{\mathit{\_}\mathit{f}}}{(-\mathit{\_}\mathit{f}{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^3+\mathit{\_}\mathit{f}y\left(x\right)-{\left(y\left(x\right)\right)}^2)}^{-\frac{3}{2}}\mathrm{d}\mathit{\_}\mathit{f}d\mathit{\_}\mathit{f}-{\left(y\left(x\right)\right)}^3{\int }^x\frac{1}{\mathit{\_}\mathit{f}-1}{\mathrm{e}}^{\int \frac{1}{\mathit{\_}\mathit{f}(\mathit{\_}\mathit{f}-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{\mathit{\_}\mathit{f}}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{\mathit{\_}\mathit{f}}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{f}}\int 1{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{\mathit{\_}\mathit{f}(\mathit{\_}\mathit{f}-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{\mathit{\_}\mathit{f}}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{\mathit{\_}\mathit{f}}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{f}}\frac{1}{\sqrt{\mathit{\_}\mathit{f}}}{(-\mathit{\_}\mathit{f}{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^3+\mathit{\_}\mathit{f}y\left(x\right)-{\left(y\left(x\right)\right)}^2)}^{-\frac{3}{2}}\mathrm{d}\mathit{\_}\mathit{f}d\mathit{\_}\mathit{f}+\frac{{\left(y\left(x\right)\right)}^2}{2}{\int }^x\frac{1}{\mathit{\_}\mathit{f}-1}{\mathrm{e}}^{\int \frac{1}{\mathit{\_}\mathit{f}(\mathit{\_}\mathit{f}-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{\mathit{\_}\mathit{f}}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{\mathit{\_}\mathit{f}}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{f}}\int 1{\mathrm{e}}^{-\frac{1}{2}\int \frac{1}{\mathit{\_}\mathit{f}(\mathit{\_}\mathit{f}-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{\mathit{\_}\mathit{f}}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{\mathit{\_}\mathit{f}}\right)\right)}^{-1}\mathrm{d}\mathit{\_}\mathit{f}}\frac{1}{\sqrt{\mathit{\_}\mathit{f}}}{(-\mathit{\_}\mathit{f}{\left(y\left(x\right)\right)}^2+{\left(y\left(x\right)\right)}^3+\mathit{\_}\mathit{f}y\left(x\right)-{\left(y\left(x\right)\right)}^2)}^{-\frac{3}{2}}\mathrm{d}\mathit{\_}\mathit{f}d\mathit{\_}\mathit{f}-{\mathrm{e}}^{\int \frac{1}{2x(x-1)}(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)x-\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)+\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right)){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}{\int }^{y\left(x\right)}\frac{1}{\sqrt{-\mathit{\_}\mathit{a}(\mathit{\_}\mathit{a}-1)(-\mathit{\_}\mathit{a}+x)}}d\mathit{\_}\mathit{a})\mathit{\_}\mathit{C}\mathit{1}+\int \frac{1}{x-1}{\mathrm{e}}^{\int \frac{1}{x(x-1)}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{E}\left(\sqrt{x}\right){\left(\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{K}\left(\sqrt{x}\right)\right)}^{-1}\mathrm{d}x}\mathrm{d}x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$