44.1.48 problem 50
Internal
problem
ID
[6923]
Book
:
A
First
Course
in
Differential
Equations
with
Modeling
Applications
by
Dennis
G.
Zill.
12
ed.
Metric
version.
2024.
Cengage
learning.
Section
:
Chapter
1.
Introduction
to
differential
equations.
Exercises
1.1
at
page
12
Problem
number
:
50
Date
solved
:
Friday, March 14, 2025 at 01:49:10 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=4 y \left (t \right )+{\mathrm e}^{t}\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=4 x \left (t \right )-{\mathrm e}^{t} \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 67
ode:=[diff(diff(x(t),t),t) = 4*y(t)+exp(t), diff(diff(y(t),t),t) = 4*x(t)-exp(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= \frac {{\mathrm e}^{t}}{5}+\cos \left (2 t \right ) c_{1} +{\mathrm e}^{-2 t} c_{2} +c_3 \,{\mathrm e}^{2 t}+c_4 \sin \left (2 t \right ) \\
y \left (t \right ) &= -\frac {{\mathrm e}^{t}}{5}-\cos \left (2 t \right ) c_{1} +{\mathrm e}^{-2 t} c_{2} +c_3 \,{\mathrm e}^{2 t}-c_4 \sin \left (2 t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.067 (sec). Leaf size: 218
ode={D[x[t],{t,2}]==4*y[t]+Exp[t],D[y[t],{t,2}]==4*x[t]-Exp[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{40} e^{-2 t} \left (8 e^{3 t}+10 c_1 e^{4 t}+5 c_2 e^{4 t}+10 c_3 e^{4 t}+5 c_4 e^{4 t}+20 (c_1-c_3) e^{2 t} \cos (2 t)+10 (c_2-c_4) e^{2 t} \sin (2 t)+10 c_1-5 c_2+10 c_3-5 c_4\right ) \\
y(t)\to \frac {1}{40} e^{-2 t} \left (-8 e^{3 t}+10 c_1 e^{4 t}+5 c_2 e^{4 t}+10 c_3 e^{4 t}+5 c_4 e^{4 t}-20 (c_1-c_3) e^{2 t} \cos (2 t)-10 (c_2-c_4) e^{2 t} \sin (2 t)+10 c_1-5 c_2+10 c_3-5 c_4\right ) \\
\end{align*}
✓ Sympy. Time used: 0.397 (sec). Leaf size: 119
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-4*y(t) - exp(t) + Derivative(x(t), (t, 2)),0),Eq(-4*x(t) + exp(t) + Derivative(y(t), (t, 2)),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{2} + \frac {C_{2} e^{2 t}}{2} - \frac {C_{3} \sin {\left (2 t \right )}}{2} - \frac {C_{4} \cos {\left (2 t \right )}}{2} + \frac {e^{t} \sin ^{2}{\left (2 t \right )}}{5} + \frac {e^{t} \cos ^{2}{\left (2 t \right )}}{5}, \ y{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{2} + \frac {C_{2} e^{2 t}}{2} + \frac {C_{3} \sin {\left (2 t \right )}}{2} + \frac {C_{4} \cos {\left (2 t \right )}}{2} - \frac {e^{t} \sin ^{2}{\left (2 t \right )}}{5} - \frac {e^{t} \cos ^{2}{\left (2 t \right )}}{5}\right ]
\]