23.1.422 problem 411

Internal problem ID [5029]
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 : 411
Date solved : Tuesday, September 30, 2025 at 11:27:33 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \left (4 x^{3}+a_{1} x +a_{0} \right )^{{2}/{3}}+\left (a_{0} +a_{1} y+4 y^{3}\right )^{{2}/{3}}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 36

ode:=diff(y(x),x)*(4*x^3+a__1*x+a__0)^(2/3)+(a__0+a__1*y(x)+4*y(x)^3)^(2/3) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \int \frac {1}{\left (4 x^{3}+\operatorname {a1} x +\operatorname {a0} \right )^{{2}/{3}}}d x +\int _{}^{y}\frac {1}{\left (4 \textit {\_a}^{3}+\textit {\_a} \operatorname {a1} +\operatorname {a0} \right )^{{2}/{3}}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 11.087 (sec). Leaf size: 558

ode=D[y[x],x]*(a0+a1*x+4*x^3)^(2/3)+(a0+a1*y[x]+4*y[x]^3)^(2/3)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.500 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
y = Function("y") 
ode = Eq((a0 + a1*x + 4*x**3)**(2/3)*Derivative(y(x), x) + (a0 + a1*y(x) + 4*y(x)**3)**(2/3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\left (4 y^{3} + y a_{1} + a_{0}\right )^{\frac {2}{3}}}\, dy = C_{1} - \int \frac {1}{\left (a_{0} + a_{1} x + 4 x^{3}\right )^{\frac {2}{3}}}\, dx \]