Internal problem ID [9200]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 866.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\[ \boxed {y^{\prime }-\sqrt {x^{2}+2 a x +a^{2}+4 y}-x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}-\sqrt {x^{2}+2 a x +a^{2}+4 y}\, x^{3}=-\frac {x}{2}-\frac {a}{2}} \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 37
dsolve(diff(y(x),x) = -1/2*x-1/2*a+(x^2+2*a*x+a^2+4*y(x))^(1/2)+x^2*(x^2+2*a*x+a^2+4*y(x))^(1/2)+x^3*(x^2+2*a*x+a^2+4*y(x))^(1/2),y(x), singsol=all)
\[ c_{1} +\frac {x^{4}}{2}+\frac {2 x^{3}}{3}+2 x -\sqrt {x^{2}+2 a x +a^{2}+4 y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.784 (sec). Leaf size: 85
DSolve[y'[x] == -1/2*a - x/2 + Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]] + x^2*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]] + x^3*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {a^2}{4}-\frac {a x}{2}+\frac {x^8}{16}+\frac {x^7}{6}+\frac {x^6}{9}+\frac {x^5}{2}-\frac {1}{6} (-4+3 c_1) x^4-\frac {2 c_1 x^3}{3}+\frac {3 x^2}{4}-2 c_1 x+c_1{}^2 \]