60.2.11 problem 587

Internal problem ID [10585]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 587
Date solved : Friday, March 14, 2025 at 02:13:48 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=\frac {\left (x^{{3}/{2}}+2 F \left (y-\frac {x^{3}}{6}\right )\right ) \sqrt {x}}{2} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 29

ode:=diff(y(x),x) = 1/2*(x^(3/2)+2*F(y(x)-1/6*x^3))*x^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \int _{\textit {\_b}}^{y}\frac {1}{F \left (\textit {\_a} -\frac {x^{3}}{6}\right )}d \textit {\_a} -\frac {2 x^{{3}/{2}}}{3}-c_{1} = 0 \]

✓ Mathematica. Time used: 0.241 (sec). Leaf size: 123

ode=D[y[x],x] == (Sqrt[x]*(x^(3/2) + 2*F[-1/6*x^3 + y[x]]))/2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (K[2]-\frac {x^3}{6}\right ) \int _1^x-\frac {K[1]^2 F''\left (K[2]-\frac {K[1]^3}{6}\right )}{2 F\left (K[2]-\frac {K[1]^3}{6}\right )^2}dK[1]+1}{F\left (K[2]-\frac {x^3}{6}\right )}dK[2]+\int _1^x\left (\frac {K[1]^2}{2 F\left (y(x)-\frac {K[1]^3}{6}\right )}+\sqrt {K[1]}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
F = Function("F") 
ode = Eq(-sqrt(x)*(x**(3/2) + 2*F(-x**3/6 + y(x)))/2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(x)*F(-x**3/6 + y(x)) - x**2/2 + Derivative(y(x), x) cannot be solved by the factorable group method