2.1691   ODE No. 1691

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

$$y^{″}(x){(a{x}^2+bx+c)}^{3/2}-f\left(\frac{y(x)}{\sqrt{a{x}^2+bx+c}}\right)=0$$ ✗ Mathematica : cpu = 60.6351 (sec), leaf count = 0 , could not solve

DSolve[-f[y[x]/Sqrt[c + b*x + a*x^2]] + (c + b*x + a*x^2)^(3/2)*Derivative[2][y][x] == 0, y[x], x]

✓ Maple : cpu = 1.861 (sec), leaf count = 254

$$\{y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-2{\int }^{\mathit{\_}\mathit{Z}}\frac{a}{\sqrt{4\mathit{\_}\mathit{C}\mathit{1}{a}^2-4c{\mathit{\_}\mathit{g}}^{2}a+b^2{\mathit{\_}\mathit{g}}^2+8\int F\left(\mathit{\_}\mathit{g}\right)\mathrm{d}\mathit{\_}\mathit{g}}}d\mathit{\_}\mathit{g}\sqrt{4ac-b^2}+\mathit{\_}\mathit{C}\mathit{2}\sqrt{4ac-b^2}-2\arctan \left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right)a)\sqrt{a{x}^2+bx+c},y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(2{\int }^{\mathit{\_}\mathit{Z}}\frac{a}{\sqrt{4\mathit{\_}\mathit{C}\mathit{1}{a}^2-4c{\mathit{\_}\mathit{g}}^{2}a+b^2{\mathit{\_}\mathit{g}}^2+8\int F\left(\mathit{\_}\mathit{g}\right)\mathrm{d}\mathit{\_}\mathit{g}}}d\mathit{\_}\mathit{g}\sqrt{4ac-b^2}+\mathit{\_}\mathit{C}\mathit{2}\sqrt{4ac-b^2}-2\arctan \left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right)a)\sqrt{a{x}^2+bx+c},y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(4\mathit{\_}\mathit{Z}ac-\mathit{\_}\mathit{Z}{b}^2-4F\left(\frac{\mathit{\_}\mathit{Z}}{\sqrt{a{x}^2+bx+c}}\right)\sqrt{a{x}^2+bx+c})\}$$