Internal problem ID [8230]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 650.
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}-x \sqrt {x^{2}+2 a x +a^{2}+4 y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.484 (sec). Leaf size: 35
dsolve(diff(y(x),x) = -1/2*x-1/2*a+x*(x^2+2*a*x+a^2+4*y(x))^(1/2),y(x), singsol=all)
\[ c_{1}+\frac {a^{2}}{4}+\frac {x^{2}}{4}+\frac {1}{16}-\frac {\sqrt {x^{2}+2 x a +a^{2}+4 y \relax (x )}}{4} = 0 \]
✓ Solution by Mathematica
Time used: 0.666 (sec). Leaf size: 31
DSolve[y'[x] == -1/2*a - x/2 + x*Sqrt[a^2 + 2*a*x + x^2 + 4*y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{4} \left (a+x^2+x-2 c_1\right ) \left (a-x^2+x+2 c_1\right ) \\ \end{align*}