29.34.14 problem 1016
Internal
problem
ID
[5585]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1016
Date
solved
:
Sunday, March 30, 2025 at 09:05:03 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}&=a \,x^{n} \end{align*}
✓ Maple. Time used: 0.043 (sec). Leaf size: 98
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: 0.008 (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: 1.193 (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)
\[
\left [ y{\left (x \right )} = \frac {C_{1} \left (n + 3\right ) + 3 \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}}}{n + 3}, \ y{\left (x \right )} = \frac {C_{1} \left (n + 3\right ) - 3 \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}} - 3 \sqrt {3} i \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}}}{2 \left (n + 3\right )}, \ y{\left (x \right )} = \frac {C_{1} \left (n + 3\right ) - 3 \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}} + 3 \sqrt {3} i \sqrt [3]{a} x e^{\frac {n \log {\left (x \right )}}{3}}}{2 \left (n + 3\right )}\right ]
\]