2.7.5 Problem 5
Internal
problem
ID
[19754]
Book
:
Elementary
Differential
Equations.
By
Thornton
C.
Fry.
D
Van
Nostrand.
NY.
First
Edition
(1929)
Section
:
Chapter
VII.
Linear
equations
of
order
higher
than
the
first.
section
56.
Problems
at
page
163
Problem
number
:
5
Date
solved
:
Thursday, December 11, 2025 at 01:54:21 PM
CAS
classification
:
[[_high_order, _missing_x]]
Entering higher order ode solver
\begin{align*}
y^{\prime \prime \prime \prime }-a^{4} y&=0 \\
\end{align*}
2.7.5.1 Solved as higher order constant coeff ode
0.033 (sec)
The characteristic equation is
\[ -a^{4}+\lambda ^{4} = 0 \]
The roots of the above equation are \begin{align*} \lambda _1 &= a\\ \lambda _2 &= -a\\ \lambda _3 &= i a\\ \lambda _4 &= -i a \end{align*}
Therefore the homogeneous solution is
\[ y_h(x)={\mathrm e}^{a x} c_1 +{\mathrm e}^{i a x} c_2 +{\mathrm e}^{-a x} c_3 +{\mathrm e}^{-i a x} c_4 \]
The fundamental set of solutions for the homogeneous
solution are the following \begin{align*} y_1 &= {\mathrm e}^{a x}\\ y_2 &= {\mathrm e}^{i a x}\\ y_3 &= {\mathrm e}^{-a x}\\ y_4 &= {\mathrm e}^{-i a x} \end{align*}
2.7.5.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 30
ode:=diff(diff(diff(diff(y(x),x),x),x),x)-a^4*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 \,{\mathrm e}^{-a x}+c_2 \,{\mathrm e}^{a x}+c_3 \sin \left (a x \right )+c_4 \cos \left (a x \right )
\]
Maple trace
Methods for high order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
<- constant coefficients successful
2.7.5.3 ✓ Mathematica. Time used: 0.002 (sec). Leaf size: 53
ode=D[y[x],{x,4}]-a^2*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_2 e^{-\sqrt {a} x}+c_4 e^{\sqrt {a} x}+c_1 \cos \left (\sqrt {a} x\right )+c_3 \sin \left (\sqrt {a} x\right ) \end{align*}
2.7.5.4 ✓ Sympy. Time used: 0.076 (sec). Leaf size: 32
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**4*y(x) + Derivative(y(x), (x, 4)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{- a x} + C_{2} e^{a x} + C_{3} e^{- i a x} + C_{4} e^{i a x}
\]