Internal problem ID [9192]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 857.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\[ \boxed {y^{\prime }-\sqrt {x^{2}-2 x +1+8 y}-x^{2} \sqrt {x^{2}-2 x +1+8 y}-\sqrt {x^{2}-2 x +1+8 y}\, x^{3}=-\frac {x}{4}+\frac {1}{4}} \]
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 32
dsolve(diff(y(x),x) = -1/4*x+1/4+(x^2-2*x+1+8*y(x))^(1/2)+x^2*(x^2-2*x+1+8*y(x))^(1/2)+x^3*(x^2-2*x+1+8*y(x))^(1/2),y(x), singsol=all)
\[ c_{1} +x^{4}+\frac {4 x^{3}}{3}+4 x -\sqrt {x^{2}-2 x +1+8 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.798 (sec). Leaf size: 77
DSolve[y'[x] == 1/4 - x/4 + Sqrt[1 - 2*x + x^2 + 8*y[x]] + x^2*Sqrt[1 - 2*x + x^2 + 8*y[x]] + x^3*Sqrt[1 - 2*x + x^2 + 8*y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^8}{8}+\frac {x^7}{3}+\frac {2 x^6}{9}+x^5+\left (\frac {4}{3}-c_1\right ) x^4-\frac {4 c_1 x^3}{3}+\frac {15 x^2}{8}+\left (\frac {1}{4}-4 c_1\right ) x-\frac {1}{8}+2 c_1{}^2 \]