23.5.157 problem 157
Internal
problem
ID
[6766]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
5.
THE
EQUATION
IS
LINEAR
AND
OF
ORDER
GREATER
THAN
TWO,
page
410
Problem
number
:
157
Date
solved
:
Tuesday, September 30, 2025 at 03:51:30 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} -y^{\prime }+y^{\prime \prime }-3 y^{\prime \prime \prime }+y^{\prime \prime \prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 184
ode:=-diff(y(x),x)+diff(diff(y(x),x),x)-3*diff(diff(diff(y(x),x),x),x)+diff(diff(diff(diff(y(x),x),x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 +c_2 \,{\mathrm e}^{\frac {\left (\left (27+3 \sqrt {57}\right )^{{2}/{3}}+3 \left (27+3 \sqrt {57}\right )^{{1}/{3}}+6\right ) x}{3 \left (27+3 \sqrt {57}\right )^{{1}/{3}}}}-c_3 \,{\mathrm e}^{-\frac {\left (6+\left (27+3 \sqrt {57}\right )^{{2}/{3}}-6 \left (27+3 \sqrt {57}\right )^{{1}/{3}}\right ) x}{6 \left (27+3 \sqrt {57}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (27+3 \sqrt {3}\, \sqrt {19}\right )^{{2}/{3}}-6\right ) x}{6 \left (27+3 \sqrt {3}\, \sqrt {19}\right )^{{1}/{3}}}\right )+c_4 \,{\mathrm e}^{-\frac {\left (6+\left (27+3 \sqrt {57}\right )^{{2}/{3}}-6 \left (27+3 \sqrt {57}\right )^{{1}/{3}}\right ) x}{6 \left (27+3 \sqrt {57}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (27+3 \sqrt {3}\, \sqrt {19}\right )^{{2}/{3}}-6\right ) x}{6 \left (27+3 \sqrt {3}\, \sqrt {19}\right )^{{1}/{3}}}\right )
\]
✓ Mathematica. Time used: 0.021 (sec). Leaf size: 143
ode=-D[y[x],x] + D[y[x],{x,2}] - 3*D[y[x],{x,3}] + D[y[x],{x,4}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]\right )}{\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+\text {$\#$1}-1\&,3\right ]}+\frac {c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]\right )}{\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+\text {$\#$1}-1\&,2\right ]}+\frac {c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]\right )}{\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+\text {$\#$1}-1\&,1\right ]}+c_4 \end{align*}
✓ Sympy. Time used: 0.333 (sec). Leaf size: 184
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 3*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + C_{2} e^{x \left (- \frac {\sqrt [3]{\frac {\sqrt {57}}{9} + 1}}{2} - \frac {1}{3 \sqrt [3]{\frac {\sqrt {57}}{9} + 1}} + 1\right )} \sin {\left (\frac {\sqrt {3} x \left (- 3 \sqrt [3]{\frac {\sqrt {57}}{9} + 1} + \frac {2}{\sqrt [3]{\frac {\sqrt {57}}{9} + 1}}\right )}{6} \right )} + C_{3} e^{x \left (- \frac {\sqrt [3]{\frac {\sqrt {57}}{9} + 1}}{2} - \frac {1}{3 \sqrt [3]{\frac {\sqrt {57}}{9} + 1}} + 1\right )} \cos {\left (\frac {\sqrt {3} x \left (- 3 \sqrt [3]{\frac {\sqrt {57}}{9} + 1} + \frac {2}{\sqrt [3]{\frac {\sqrt {57}}{9} + 1}}\right )}{6} \right )} + C_{4} e^{x \left (\frac {2}{3 \sqrt [3]{\frac {\sqrt {57}}{9} + 1}} + 1 + \sqrt [3]{\frac {\sqrt {57}}{9} + 1}\right )}
\]