82.48.19 problem Ex. 19
Internal
problem
ID
[18929]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
VIII.
End
of
chapter
problems
at
page
107
Problem
number
:
Ex.
19
Date
solved
:
Monday, March 31, 2025 at 06:25:16 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y^{3} y^{\prime \prime }&=a \end{align*}
✓ Maple. Time used: 0.228 (sec). Leaf size: 46
ode:=y(x)^3*diff(diff(y(x),x),x) = a;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\sqrt {\left (\left (c_2 +x \right )^{2} c_1^{2}+a \right ) c_1}}{c_1} \\
y &= -\frac {\sqrt {\left (\left (c_2 +x \right )^{2} c_1^{2}+a \right ) c_1}}{c_1} \\
\end{align*}
✓ Mathematica. Time used: 2.776 (sec). Leaf size: 63
ode=y[x]^3*D[y[x],{x,2}]==a;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}} \\
y(x)\to \frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}} \\
y(x)\to \text {Indeterminate} \\
\end{align*}
✓ Sympy. Time used: 0.562 (sec). Leaf size: 112
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a + y(x)**3*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} \frac {i \sqrt {-1 + \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \left |{\frac {a}{C_{1} y^{2}{\left (x \right )}}}\right | > 1 \\\frac {\sqrt {1 - \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} \frac {i \sqrt {-1 + \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \left |{\frac {a}{C_{1} y^{2}{\left (x \right )}}}\right | > 1 \\\frac {\sqrt {1 - \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} - x\right ]
\]