54.1.39 problem 39

Internal problem ID [11353]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 08:03:58 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }-\operatorname {a3} y^{3}-\operatorname {a2} y^{2}-\operatorname {a1} y-\operatorname {a0}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

ode:=diff(y(x),x)-a3*y(x)^3-a2*y(x)^2-a1*y(x)-a0 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ x -\int _{}^{y}\frac {1}{\textit {\_a}^{3} \operatorname {a3} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.174 (sec). Leaf size: 566

ode=D[y[x],x] - a3*y[x]^3 - a2*y[x]^2 - a1*y[x] - a0==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 41.911 (sec). Leaf size: 274

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(-a0 - a1*y(x) - a2*y(x)**2 - a3*y(x)**3 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \operatorname {RootSum} {\left (t^{3} \left (27 a_{0}^{2} a_{3}^{2} - 18 a_{0} a_{1} a_{2} a_{3} + 4 a_{0} a_{2}^{3} + 4 a_{1}^{3} a_{3} - a_{1}^{2} a_{2}^{2}\right ) + t \left (- 3 a_{1} a_{3} + a_{2}^{2}\right ) - a_{3}, \left ( t \mapsto t \log {\left (y{\left (x \right )} + \frac {- 162 t^{2} a_{0}^{2} a_{1} a_{3}^{3} + 54 t^{2} a_{0}^{2} a_{2}^{2} a_{3}^{2} + 108 t^{2} a_{0} a_{1}^{2} a_{2} a_{3}^{2} - 60 t^{2} a_{0} a_{1} a_{2}^{3} a_{3} + 8 t^{2} a_{0} a_{2}^{5} - 24 t^{2} a_{1}^{4} a_{3}^{2} + 14 t^{2} a_{1}^{3} a_{2}^{2} a_{3} - 2 t^{2} a_{1}^{2} a_{2}^{4} + 81 t a_{0}^{2} a_{3}^{3} - 54 t a_{0} a_{1} a_{2} a_{3}^{2} + 12 t a_{0} a_{2}^{3} a_{3} + 12 t a_{1}^{3} a_{3}^{2} - 3 t a_{1}^{2} a_{2}^{2} a_{3} + 9 a_{0} a_{2} a_{3}^{2} + 12 a_{1}^{2} a_{3}^{2} - 11 a_{1} a_{2}^{2} a_{3} + 2 a_{2}^{4}}{a_{3} \left (27 a_{0} a_{3}^{2} - 9 a_{1} a_{2} a_{3} + 2 a_{2}^{3}\right )} \right )} \right )\right )} = C_{1} - x \]