26.7.21 problem Exercise 20.22, page 220
Internal
problem
ID
[7071]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
4.
Higher
order
linear
differential
equations.
Lesson
20.
Constant
coefficients
Problem
number
:
Exercise
20.22,
page
220
Date
solved
:
Tuesday, September 30, 2025 at 04:21:07 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime \prime }+36 y&=0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 48
ode:=diff(diff(diff(diff(y(x),x),x),x),x)-8*diff(diff(y(x),x),x)+36*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 \,{\mathrm e}^{\sqrt {5}\, x} \sin \left (x \right )-c_2 \,{\mathrm e}^{-\sqrt {5}\, x} \sin \left (x \right )+c_3 \,{\mathrm e}^{\sqrt {5}\, x} \cos \left (x \right )+c_4 \,{\mathrm e}^{-\sqrt {5}\, x} \cos \left (x \right )
\]
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 142
ode=D[y[x],{x,4}]-8*D[y[x],{x,2}]+36*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-\sqrt {6} x \cos \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )} \left (\left (c_3 e^{2 \sqrt {6} x \cos \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )}+c_2\right ) \cos \left (\sqrt {6} x \sin \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )\right )+\sin \left (\sqrt {6} x \sin \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )\right ) \left (c_1 e^{2 \sqrt {6} x \cos \left (\frac {1}{2} \arctan \left (\frac {\sqrt {5}}{2}\right )\right )}+c_4\right )\right ) \end{align*}
✓ Sympy. Time used: 0.153 (sec). Leaf size: 131
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(36*y(x) - 8*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{1} \sin {\left (\sqrt {6} x \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {5}}{2} \right )}}{2} \right )} \right )} + C_{2} \cos {\left (\sqrt {6} x \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {5}}{2} \right )}}{2} \right )} \right )}\right ) e^{- \sqrt {6} x \cos {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {5}}{2} \right )}}{2} \right )}} + \left (C_{3} \sin {\left (\sqrt {6} x \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {5}}{2} \right )}}{2} \right )} \right )} + C_{4} \cos {\left (\sqrt {6} x \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {5}}{2} \right )}}{2} \right )} \right )}\right ) e^{\sqrt {6} x \cos {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {5}}{2} \right )}}{2} \right )}}
\]