30.12.36 problem 45 (b)
Internal
problem
ID
[7628]
Book
:
Fundamentals
of
Differential
Equations.
By
Nagle,
Saff
and
Snider.
9th
edition.
Boston.
Pearson
2018.
Section
:
Chapter
4,
Linear
Second-Order
Equations.
EXERCISES
4.2
at
page
164
Problem
number
:
45
(b)
Date
solved
:
Tuesday, September 30, 2025 at 04:55:06 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime }+5 y&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 65
ode:=diff(diff(diff(diff(y(t),t),t),t),t)-5*diff(diff(y(t),t),t)+5*y(t) = 0;
dsolve(ode,y(t), singsol=all);
\[
y = c_1 \,{\mathrm e}^{-\frac {\sqrt {10+2 \sqrt {5}}\, t}{2}}+c_2 \,{\mathrm e}^{\frac {\sqrt {10+2 \sqrt {5}}\, t}{2}}+c_3 \,{\mathrm e}^{-\frac {\sqrt {10-2 \sqrt {5}}\, t}{2}}+c_4 \,{\mathrm e}^{\frac {\sqrt {10-2 \sqrt {5}}\, t}{2}}
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 100
ode=D[y[t],{t,4}]-5*D[y[t],{t,2}]+5*y[t]==0;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to c_1 e^{\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} t}+c_2 e^{-\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} t}+c_3 e^{\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} t}+c_4 e^{-\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} t} \end{align*}
✓ Sympy. Time used: 0.163 (sec). Leaf size: 172
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(5*y(t) - 5*Derivative(y(t), (t, 2)) + 3*Derivative(y(t), (t, 4)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (C_{1} \sin {\left (\frac {3^{\frac {3}{4}} \sqrt [4]{5} t \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {35}}{5} \right )}}{2} \right )}}{3} \right )} + C_{2} \cos {\left (\frac {3^{\frac {3}{4}} \sqrt [4]{5} t \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {35}}{5} \right )}}{2} \right )}}{3} \right )}\right ) e^{- \frac {3^{\frac {3}{4}} \sqrt [4]{5} t \cos {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {35}}{5} \right )}}{2} \right )}}{3}} + \left (C_{3} \sin {\left (\frac {3^{\frac {3}{4}} \sqrt [4]{5} t \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {35}}{5} \right )}}{2} \right )}}{3} \right )} + C_{4} \cos {\left (\frac {3^{\frac {3}{4}} \sqrt [4]{5} t \sin {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {35}}{5} \right )}}{2} \right )}}{3} \right )}\right ) e^{\frac {3^{\frac {3}{4}} \sqrt [4]{5} t \cos {\left (\frac {\operatorname {atan}{\left (\frac {\sqrt {35}}{5} \right )}}{2} \right )}}{3}}
\]