Internal problem ID [8224]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 644.
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} \left (a x -2 \sqrt {a \left (a \,x^{4}+8 y\right )}\right )}{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 27
dsolve(diff(y(x),x) = -1/2*x^2*(a*x-2*(a*(a*x^4+8*y(x)))^(1/2)),y(x), singsol=all)
\[ c_{1}+\frac {4 a \,x^{3}}{3}-\sqrt {a \left (a \,x^{4}+8 y \relax (x )\right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.511 (sec). Leaf size: 34
DSolve[y'[x] == -1/2*(x^2*(a*x - 2*Sqrt[a*(a*x^4 + 8*y[x])])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{72} a \left (16 x^6-9 x^4-96 c_1 x^3+144 c_1{}^2\right ) \\ \end{align*}