82.22.2 problem Ex. 2
Internal
problem
ID
[18777]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
IV.
Singular
solutions.
problems
at
page
48
Problem
number
:
Ex.
2
Date
solved
:
Thursday, March 13, 2025 at 12:56:00 PM
CAS
classification
:
[_quadrature]
\begin{align*} a {y^{\prime }}^{3}&=27 y \end{align*}
✓ Maple. Time used: 0.023 (sec). Leaf size: 191
ode:=a*diff(y(x),x)^3 = 27*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= 0 \\
y \left (x \right ) &= -\frac {2 \left (-x +c_{1} \right ) \sqrt {2}\, \sqrt {\left (-c_{1} +x \right ) a}}{a} \\
y \left (x \right ) &= \frac {2 \left (-x +c_{1} \right ) \sqrt {2}\, \sqrt {\left (-c_{1} +x \right ) a}}{a} \\
y \left (x \right ) &= \frac {\left (i-\sqrt {3}\right ) \sqrt {a \left (-c_{1} +x \right ) \left (1+i \sqrt {3}\right )}\, \left (-c_{1} +x \right )}{a} \\
y \left (x \right ) &= \frac {\left (-x +c_{1} \right ) \left (i-\sqrt {3}\right ) \sqrt {a \left (-c_{1} +x \right ) \left (1+i \sqrt {3}\right )}}{a} \\
y \left (x \right ) &= \frac {\left (-x +c_{1} \right ) \left (\sqrt {3}+i\right ) \sqrt {\left (i \sqrt {3}-1\right ) \left (-x +c_{1} \right ) a}}{a} \\
y \left (x \right ) &= \frac {\left (\sqrt {3}+i\right ) \sqrt {\left (i \sqrt {3}-1\right ) \left (-x +c_{1} \right ) a}\, \left (-c_{1} +x \right )}{a} \\
\end{align*}
✓ Mathematica. Time used: 14.355 (sec). Leaf size: 106
ode=a*D[y[x],x]^3==27*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {2}{3} \sqrt {\frac {2}{3}} \left (\frac {3 x}{\sqrt [3]{a}}+c_1\right ){}^{3/2} \\
y(x)\to \frac {2}{3} \sqrt {\frac {2}{3}} \left (-\frac {3 \sqrt [3]{-1} x}{\sqrt [3]{a}}+c_1\right ){}^{3/2} \\
y(x)\to \frac {2}{3} \sqrt {\frac {2}{3}} \left (\frac {3 (-1)^{2/3} x}{\sqrt [3]{a}}+c_1\right ){}^{3/2} \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 96.722 (sec). Leaf size: 236
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*Derivative(y(x), x)**3 - 27*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {2 \sqrt {6} \sqrt {\frac {C_{1} + 3 x}{a}} \left (C_{1} + 3 x\right )}{9}, \ y{\left (x \right )} = \frac {2 \sqrt {6} \sqrt {\frac {C_{1} + 3 x}{a}} \left (C_{1} + 3 x\right )}{9}, \ y{\left (x \right )} = \sqrt {\frac {2 C_{1} - 3 x - 3 \sqrt {3} i x}{a}} \left (- \frac {2 \sqrt {3} C_{1}}{9} + \frac {\sqrt {3} x}{3} + i x\right ), \ y{\left (x \right )} = \sqrt {\frac {2 C_{1} - 3 x - 3 \sqrt {3} i x}{a}} \left (\frac {2 \sqrt {3} C_{1}}{9} - \frac {\sqrt {3} x}{3} - i x\right ), \ y{\left (x \right )} = \sqrt {\frac {2 C_{1} - 3 x + 3 \sqrt {3} i x}{a}} \left (- \frac {2 \sqrt {3} C_{1}}{9} + \frac {\sqrt {3} x}{3} - i x\right ), \ y{\left (x \right )} = \sqrt {\frac {2 C_{1} - 3 x + 3 \sqrt {3} i x}{a}} \left (\frac {2 \sqrt {3} C_{1}}{9} - \frac {\sqrt {3} x}{3} + i x\right )\right ]
\]