90.18.6 problem 10
Internal
problem
ID
[25278]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
4.
Linear
Constant
Coefficient
Differential
Equations.
Exercises
at
page
299
Problem
number
:
10
Date
solved
:
Thursday, October 02, 2025 at 11:59:32 PM
CAS
classification
:
[[_high_order, _missing_y]]
\begin{align*} y^{\prime \prime \prime \prime }-4 y^{\prime \prime }+4 y^{\prime }&=4 \,{\mathrm e}^{2 t} t \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 221
ode:=diff(diff(diff(diff(y(t),t),t),t),t)-4*diff(diff(y(t),t),t)+4*diff(y(t),t) = 4*exp(2*t)*t;
dsolve(ode,y(t), singsol=all);
\[
y = \int \left ({\mathrm e}^{-\frac {\left (54+6 \sqrt {33}\right )^{{1}/{3}} t \left (-24+\left (\sqrt {33}-9\right ) \left (54+6 \sqrt {33}\right )^{{1}/{3}}\right )}{144}} \cos \left (\frac {\left (54+6 \sqrt {3}\, \sqrt {11}\right )^{{1}/{3}} \sqrt {3}\, \left (\left (54+6 \sqrt {3}\, \sqrt {11}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {11}-9 \left (54+6 \sqrt {3}\, \sqrt {11}\right )^{{1}/{3}}+24\right ) t}{144}\right ) c_2 +{\mathrm e}^{-\frac {\left (54+6 \sqrt {33}\right )^{{1}/{3}} t \left (-24+\left (\sqrt {33}-9\right ) \left (54+6 \sqrt {33}\right )^{{1}/{3}}\right )}{144}} \sin \left (\frac {\left (54+6 \sqrt {3}\, \sqrt {11}\right )^{{1}/{3}} \sqrt {3}\, \left (\left (54+6 \sqrt {3}\, \sqrt {11}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {11}-9 \left (54+6 \sqrt {3}\, \sqrt {11}\right )^{{1}/{3}}+24\right ) t}{144}\right ) c_3 +c_1 \,{\mathrm e}^{\frac {\left (54+6 \sqrt {33}\right )^{{1}/{3}} t \left (-24+\left (\sqrt {33}-9\right ) \left (54+6 \sqrt {33}\right )^{{1}/{3}}\right )}{72}}+{\mathrm e}^{2 t} \left (t -2\right )\right )d t +c_4
\]
✓ Mathematica. Time used: 0.049 (sec). Leaf size: 134
ode=D[y[t],{t,4}]-4*D[y[t],{t,2}]+4*D[y[t],{t,1}]==4*t*Exp[2*t];
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3-4 \text {$\#$1}+4\&,3\right ]\right )}{\text {Root}\left [\text {$\#$1}^3-4 \text {$\#$1}+4\&,3\right ]}+\frac {c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3-4 \text {$\#$1}+4\&,2\right ]\right )}{\text {Root}\left [\text {$\#$1}^3-4 \text {$\#$1}+4\&,2\right ]}+\frac {c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3-4 \text {$\#$1}+4\&,1\right ]\right )}{\text {Root}\left [\text {$\#$1}^3-4 \text {$\#$1}+4\&,1\right ]}+e^{2 t} \left (\frac {t}{2}-\frac {5}{4}\right )+c_4 \end{align*}
✓ Sympy. Time used: 0.436 (sec). Leaf size: 230
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-4*t*exp(2*t) + 4*Derivative(y(t), t) - 4*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = C_{1} + C_{2} e^{\frac {\sqrt [3]{6} t \left (\frac {2 \sqrt [3]{6}}{\sqrt [3]{\sqrt {33} + 9}} + \sqrt [3]{\sqrt {33} + 9}\right )}{6}} \sin {\left (\sqrt [3]{2} \sqrt [6]{3} t \left (- \frac {3^{\frac {2}{3}} \sqrt [3]{\sqrt {33} + 9}}{6} + \frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {33} + 9}}\right ) \right )} + C_{3} e^{\frac {\sqrt [3]{6} t \left (\frac {2 \sqrt [3]{6}}{\sqrt [3]{\sqrt {33} + 9}} + \sqrt [3]{\sqrt {33} + 9}\right )}{6}} \cos {\left (\sqrt [3]{2} \sqrt [6]{3} t \left (- \frac {3^{\frac {2}{3}} \sqrt [3]{\sqrt {33} + 9}}{6} + \frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {33} + 9}}\right ) \right )} + C_{4} e^{- \frac {\sqrt [3]{6} t \left (\frac {2 \sqrt [3]{6}}{\sqrt [3]{\sqrt {33} + 9}} + \sqrt [3]{\sqrt {33} + 9}\right )}{3}} + \frac {t e^{2 t}}{2} - \frac {5 e^{2 t}}{4}
\]