Internal problem ID [8163]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 583.
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 {\left (x^{2} a -2 F \left (y+\frac {x^{4} a}{8}\right )\right ) x}{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.039 (sec). Leaf size: 44
dsolve(diff(y(x),x) = -1/2*(a*x^2-2*F(y(x)+1/8*a*x^4))*x,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {a \,x^{4}}{8}+\RootOf \left (F \left (\textit {\_Z} \right )\right ) \\ y \relax (x ) = -\frac {a \,x^{4}}{8}+\RootOf \left (-x^{2}+2 \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right )+2 c_{1}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.19 (sec). Leaf size: 126
DSolve[y'[x] == -1/2*(x*(a*x^2 - 2*F[(a*x^4)/8 + y[x]])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {a x^4}{8}+K[2]\right ) \int _1^x\frac {a K[1]^3 F'\left (\frac {1}{8} a K[1]^4+K[2]\right )}{2 F\left (\frac {1}{8} a K[1]^4+K[2]\right )^2}dK[1]+1}{F\left (\frac {a x^4}{8}+K[2]\right )}dK[2]+\int _1^x\left (K[1]-\frac {a K[1]^3}{2 F\left (\frac {1}{8} a K[1]^4+y(x)\right )}\right )dK[1]=c_1,y(x)\right ] \]