23.4.54 problem 54
Internal
problem
ID
[6356]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
54
Date
solved
:
Tuesday, September 30, 2025 at 02:52:26 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]
\begin{align*} a {y^{\prime }}^{3}+y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 40
ode:=a*diff(y(x),x)^3+diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (c_1 +x \right ) \sqrt {2}}{\sqrt {a \left (c_1 +x \right )}}+c_2 \\
y &= -\frac {\left (c_1 +x \right ) \sqrt {2}}{\sqrt {a \left (c_1 +x \right )}}+c_2 \\
\end{align*}
✓ Mathematica. Time used: 0.203 (sec). Leaf size: 50
ode=a*D[y[x],x]^3 + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_2-\frac {\sqrt {2 a x-2 c_1}}{a}\\ y(x)&\to \frac {\sqrt {2 a x-2 c_1}}{a}+c_2 \end{align*}
✓ Sympy. Time used: 27.429 (sec). Leaf size: 316
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*Derivative(y(x), x)**3 + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \left (\begin {cases} - \frac {2 C_{1} \sqrt {- \frac {1}{C_{1} - a x}}}{a} + 2 x \sqrt {- \frac {1}{C_{1} - a x}} & \text {for}\: a \neq 0 \\x \sqrt {- \frac {1}{C_{1}}} & \text {otherwise} \end {cases}\right )}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \left (\begin {cases} - \frac {2 C_{1} \sqrt {- \frac {1}{C_{1} - a x}}}{a} + 2 x \sqrt {- \frac {1}{C_{1} - a x}} & \text {for}\: a \neq 0 \\x \sqrt {- \frac {1}{C_{1}}} & \text {otherwise} \end {cases}\right )}{2}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \left (\begin {cases} - \frac {2 C_{1} \sqrt {- \frac {1}{C_{1} - a x}}}{a} + 2 x \sqrt {- \frac {1}{C_{1} - a x}} & \text {for}\: a \neq 0 \\x \sqrt {- \frac {1}{C_{1}}} & \text {otherwise} \end {cases}\right )}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \left (\begin {cases} - \frac {2 C_{1} \sqrt {- \frac {1}{C_{1} - a x}}}{a} + 2 x \sqrt {- \frac {1}{C_{1} - a x}} & \text {for}\: a \neq 0 \\x \sqrt {- \frac {1}{C_{1}}} & \text {otherwise} \end {cases}\right )}{2}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt {2} \left (\begin {cases} - \frac {2 C_{1} \sqrt {- \frac {1}{C_{1} - a x}}}{a} + 2 x \sqrt {- \frac {1}{C_{1} - a x}} & \text {for}\: a \neq 0 \\x \sqrt {- \frac {1}{C_{1}}} & \text {otherwise} \end {cases}\right )}{2}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {2} \left (\begin {cases} - \frac {2 C_{1} \sqrt {- \frac {1}{C_{1} - a x}}}{a} + 2 x \sqrt {- \frac {1}{C_{1} - a x}} & \text {for}\: a \neq 0 \\x \sqrt {- \frac {1}{C_{1}}} & \text {otherwise} \end {cases}\right )}{2}\right ]
\]