23.2.256 problem 271

Internal problem ID [5611]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 271
Date solved : Tuesday, September 30, 2025 at 01:12:01 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}&=a \,x^{n} \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 100

ode:=diff(y(x),x)^3 = a*x^n; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {3 x \left (a \,x^{n}\right )^{{1}/{3}}+c_1 \left (n +3\right )}{n +3} \\ y &= \frac {\left (-3 i \sqrt {3}\, x -3 x \right ) \left (a \,x^{n}\right )^{{1}/{3}}+2 c_1 \left (n +3\right )}{2 n +6} \\ y &= \frac {\left (3 i \sqrt {3}\, x -3 x \right ) \left (a \,x^{n}\right )^{{1}/{3}}+2 c_1 \left (n +3\right )}{2 n +6} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 95

ode=(D[y[x],x])^3 ==a x^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {3 \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1\\ y(x)&\to -\frac {3 \sqrt [3]{-1} \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1\\ y(x)&\to \frac {3 (-1)^{2/3} \sqrt [3]{a} x^{\frac {n}{3}+1}}{n+3}+c_1 \end{align*}

✓ Sympy. Time used: 0.779 (sec). Leaf size: 133

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a*x**n + Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {C_{1} \left (n + 3\right ) + 3 \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}}}{n + 3}, \ y{\left (x \right )} = \frac {C_{1} \left (n + 3\right ) - 3 \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}} - 3 \sqrt {3} i \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}}}{2 \left (n + 3\right )}, \ y{\left (x \right )} = \frac {C_{1} \left (n + 3\right ) - 3 \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}} + 3 \sqrt {3} i \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}}}{2 \left (n + 3\right )}\right ] \]