54.2.43 problem 619

Internal problem ID [11917]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 619
Date solved : Sunday, October 12, 2025 at 02:02:49 AM
CAS classification : [`x=_G(y,y')`]

\begin{align*} y^{\prime }&=\frac {6 y}{8 y^{4}+9 y^{3}+12 y^{2}+6 y-F \left (-\frac {y^{4}}{3}-\frac {y^{3}}{2}-y^{2}-y+x \right )} \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 81

ode:=diff(y(x),x) = 6*y(x)/(8*y(x)^4+9*y(x)^3+12*y(x)^2+6*y(x)-F(-1/3*y(x)^4-1/2*y(x)^3-y(x)^2-y(x)+x)); 
dsolve(ode,y(x), singsol=all);
 

\[ \int _{\textit {\_b}}^{y}\frac {-8 \textit {\_a}^{4}-9 \textit {\_a}^{3}-12 \textit {\_a}^{2}+F \left (-\frac {1}{3} \textit {\_a}^{4}-\frac {1}{2} \textit {\_a}^{3}-\textit {\_a}^{2}-\textit {\_a} +x \right )-6 \textit {\_a}}{F \left (-\frac {1}{3} \textit {\_a}^{4}-\frac {1}{2} \textit {\_a}^{3}-\textit {\_a}^{2}-\textit {\_a} +x \right ) \textit {\_a}}d \textit {\_a} -c_1 = 0 \]

✓ Mathematica. Time used: 0.297 (sec). Leaf size: 330

ode=D[y[x],x] == (6*y[x])/(-F[x - y[x] - y[x]^2 - y[x]^3/2 - y[x]^4/3] + 6*y[x] + 12*y[x]^2 + 9*y[x]^3 + 8*y[x]^4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {8 K[2]^3}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {9 K[2]^2}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {12 K[2]}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}-\frac {F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right ) \int _1^x-\frac {6 \left (-\frac {4}{3} K[2]^3-\frac {3 K[2]^2}{2}-2 K[2]-1\right ) F''\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+K[1]\right )}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+K[1]\right )^2}dK[1]+6}{F\left (-\frac {1}{3} K[2]^4-\frac {K[2]^3}{2}-K[2]^2-K[2]+x\right )}+\frac {1}{K[2]}\right )dK[2]+\int _1^x\frac {6}{F\left (-\frac {1}{3} y(x)^4-\frac {y(x)^3}{2}-y(x)^2-y(x)+K[1]\right )}dK[1]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
F = Function("F") 
ode = Eq(Derivative(y(x), x) - 6*y(x)/(-F(x - y(x)**4/3 - y(x)**3/2 - y(x)**2 - y(x)) + 8*y(x)**4 + 9*y(x)**3 + 12*y(x)**2 + 6*y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : multiple generators [sqrt(-32*(-23*C1 - 138*x - 52)**(1/3) - 92), sqrt(2*(-23*C1/64 - 69*x/32 - 13/16)**(1/3) - 23/8 + 75/(32*sqrt(-2*(-23*C1/64 - 69*x/32 - 13/16)**(1/3) - 23/16)))] 
No algorithms are implemented to solve equation sqrt(-2*(-23*C1/64 - 69*x/32 - 13/16)**(1/3) - 23/16)/2 + sqrt(2*(-23*C1/64 - 69*x/32 - 13/16)**(1/3) - 23/8 + 75/(32*sqrt(-2*(-23*C1/64 - 69*x/32 - 13/16)**(1/3) - 23/16)))/2 + d(x) + 3/8