Internal problem ID [8446]
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)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {x}{2}+\frac {a}{2}-\sqrt {x^{2}+2 a x +a^{2}+4 y}-x^{2} \sqrt {x^{2}+2 a x +a^{2}+4 y}-x^{3} \sqrt {x^{2}+2 a x +a^{2}+4 y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.37 (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)
\[ c_{1}+\frac {x^{4}}{2}+\frac {2 x^{3}}{3}+2 x -\sqrt {a^{2}+2 a x +x^{2}+4 y \relax (x )} = 0 \]
✓ Solution by Mathematica
Time used: 0.826 (sec). Leaf size: 52
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]
\begin{align*} y(x)\to -\frac {1}{144} \left (6 a+3 x^4+4 x^3+18 x-12 c_1\right ) \left (6 a-x \left ((3 x+4) x^2+6\right )+12 c_1\right ) \\ \end{align*}