23.1.88 problem 82

Internal problem ID [4695]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 82
Date solved : Tuesday, September 30, 2025 at 08:15:17 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=a_{0} +a_{1} y+a_{2} y^{2}+a_{3} y^{3} \end{align*}

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

ode:=diff(y(x),x) = a__0+a__1*y(x)+a__2*y(x)^2+a__3*y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ x -\int _{}^{y}\frac {1}{\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 54

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

\[ \text {Solve}\left [\text {RootSum}\left [\text {$\#$1}^3 \text {a3}+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,\frac {\log (y(x)-\text {$\#$1})}{3 \text {$\#$1}^2 \text {a3}+2 \text {$\#$1} \text {a2}+\text {a1}}\&\right ]=x+c_1,y(x)\right ] \]

✓ Sympy. Time used: 41.181 (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)
 

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