60.2.7 problem 583

Internal problem ID [10581]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 583
Date solved : Sunday, March 30, 2025 at 06:08:24 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\frac {\left (a \,x^{2}-2 F \left (y+\frac {a \,x^{4}}{8}\right )\right ) x}{2} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 44

ode:=diff(y(x),x) = -1/2*(a*x^2-2*F(y(x)+1/8*a*x^4))*x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {a \,x^{4}}{8}+\operatorname {RootOf}\left (F \left (\textit {\_Z} \right )\right ) \\ y &= -\frac {a \,x^{4}}{8}+\operatorname {RootOf}\left (-x^{2}+2 \int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +2 c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.236 (sec). Leaf size: 126

ode=D[y[x],x] == -1/2*(x*(a*x^2 - 2*F[(a*x^4)/8 + y[x]])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {a x^4}{8}+K[2]\right ) \int _1^x\frac {a K[1]^3 F''\left (\frac {1}{8} a K[1]^4+K[2]\right )}{2 F\left (\frac {1}{8} a K[1]^4+K[2]\right )^2}dK[1]+1}{F\left (\frac {a x^4}{8}+K[2]\right )}dK[2]+\int _1^x\left (K[1]-\frac {a K[1]^3}{2 F\left (\frac {1}{8} a K[1]^4+y(x)\right )}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
F = Function("F") 
ode = Eq(x*(a*x**2 - 2*F(a*x**4/8 + y(x)))/2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(-a*x**2/2 + F(a*x**4/8 + y(x))) + Derivative(y(x), x) cannot be solved by the factorable group method