85.83.12 problem 4 (c)
Internal
problem
ID
[23009]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
10.
Systems
of
differential
equations
and
their
applications.
A
Exercises
at
page
444
Problem
number
:
4
(c)
Date
solved
:
Thursday, October 02, 2025 at 09:17:29 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}r \left (t \right )&=r \left (t \right )+y \left (t \right )\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=5 r \left (t \right )-3 y \left (t \right )+t^{2} \end{align*}
✗ Maple
ode:=[diff(diff(r(t),t),t) = r(t)+y(t), diff(diff(y(t),t),t) = 5*r(t)-3*y(t)+t^2];
dsolve(ode);
\[ \text {No solution found} \]
✓ Mathematica. Time used: 0.729 (sec). Leaf size: 360
ode={D[r[t],{t,2}]==r[t]+y[t],D[y[t],{t,2}]==5*r[t]-3*y[t]+t^2};
ic={};
DSolve[{ode,ic},{r[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} r(t)&\to \frac {1}{48} e^{-\sqrt {2} t} \left (-6 e^{\sqrt {2} t} t^2-3 e^{\sqrt {2} t}+20 c_1 e^{2 \sqrt {2} t}+10 \sqrt {2} c_2 e^{2 \sqrt {2} t}+4 c_3 e^{2 \sqrt {2} t}+2 \sqrt {2} c_4 e^{2 \sqrt {2} t}+8 (c_1-c_3) e^{\sqrt {2} t} \cos (2 t)+4 (c_2-c_4) e^{\sqrt {2} t} \sin (2 t)+20 c_1-10 \sqrt {2} c_2+4 c_3-2 \sqrt {2} c_4\right )\\ y(t)&\to \frac {1}{48} e^{-\sqrt {2} t} \left (6 e^{\sqrt {2} t} t^2-9 e^{\sqrt {2} t}+20 c_1 e^{2 \sqrt {2} t}+10 \sqrt {2} c_2 e^{2 \sqrt {2} t}+4 c_3 e^{2 \sqrt {2} t}+2 \sqrt {2} c_4 e^{2 \sqrt {2} t}-40 (c_1-c_3) e^{\sqrt {2} t} \cos (2 t)-20 (c_2-c_4) e^{\sqrt {2} t} \sin (2 t)+20 c_1-10 \sqrt {2} c_2+4 c_3-2 \sqrt {2} c_4\right ) \end{align*}
✓ Sympy. Time used: 0.357 (sec). Leaf size: 211
from sympy import *
t = symbols("t")
r = Function("r")
y = Function("y")
ode=[Eq(-r(t) - y(t) + Derivative(r(t), (t, 2)),0),Eq(-t**2 - 5*r(t) + 3*y(t) + Derivative(y(t), (t, 2)),0)]
ics = {}
dsolve(ode,func=[r(t),y(t)],ics=ics)
\[
\left [ r{\left (t \right )} = \frac {\sqrt {2} C_{1} e^{\sqrt {2} t}}{2} - \frac {\sqrt {2} C_{2} e^{- \sqrt {2} t}}{2} - \frac {C_{3} \sin {\left (2 t \right )}}{10} - \frac {C_{4} \cos {\left (2 t \right )}}{10} - \frac {t^{2} \sin ^{2}{\left (2 t \right )}}{24} - \frac {t^{2} \cos ^{2}{\left (2 t \right )}}{24} - \frac {t^{2}}{12} + \frac {\sin ^{2}{\left (2 t \right )}}{48} + \frac {\cos ^{2}{\left (2 t \right )}}{48} - \frac {1}{12}, \ y{\left (t \right )} = \frac {\sqrt {2} C_{1} e^{\sqrt {2} t}}{2} - \frac {\sqrt {2} C_{2} e^{- \sqrt {2} t}}{2} + \frac {C_{3} \sin {\left (2 t \right )}}{2} + \frac {C_{4} \cos {\left (2 t \right )}}{2} + \frac {5 t^{2} \sin ^{2}{\left (2 t \right )}}{24} + \frac {5 t^{2} \cos ^{2}{\left (2 t \right )}}{24} - \frac {t^{2}}{12} - \frac {5 \sin ^{2}{\left (2 t \right )}}{48} - \frac {5 \cos ^{2}{\left (2 t \right )}}{48} - \frac {1}{12}\right ]
\]