44.29.13 problem 4(c)
Internal
problem
ID
[9495]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
10.
Systems
of
First-Order
Equations.
Section
A.
Drill
exercises.
Page
400
Problem
number
:
4(c)
Date
solved
:
Tuesday, September 30, 2025 at 06:19:38 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+y \left (t \right )-t +3\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-5 y \left (t \right )+t +1 \end{align*}
✓ Maple. Time used: 0.151 (sec). Leaf size: 105
ode:=[diff(x(t),t) = -4*x(t)+y(t)-t+3, diff(y(t),t) = -x(t)-5*y(t)+t+1];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 -\frac {4 t}{21}+\frac {39}{49} \\
y \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {9 t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_2}{2}+\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_2}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_1}{2}-\frac {{\mathrm e}^{-\frac {9 t}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_1}{2}-\frac {1}{147}+\frac {5 t}{21} \\
\end{align*}
✓ Mathematica. Time used: 0.93 (sec). Leaf size: 484
ode={D[x[t],t]==-4*x[t]+y[t]-t+3,D[y[t],t]==-x[t]-5*y[t]+t+1};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{3} e^{-9 t/2} \left (\left (\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )+3 \cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \int _1^t-\frac {1}{3} e^{\frac {9 K[1]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) (K[1]-3)+\sqrt {3} (K[1]+5) \sin \left (\frac {1}{2} \sqrt {3} K[1]\right )\right )dK[1]+2 \sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t\frac {1}{3} e^{\frac {9 K[2]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[2]\right ) (K[2]+1)-\sqrt {3} (K[2]-7) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )\right )dK[2]+3 c_1 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )+2 \sqrt {3} c_2 \sin \left (\frac {\sqrt {3} t}{2}\right )\right )\\ y(t)&\to \frac {1}{3} e^{-9 t/2} \left (-2 \sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right ) \int _1^t-\frac {1}{3} e^{\frac {9 K[1]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[1]\right ) (K[1]-3)+\sqrt {3} (K[1]+5) \sin \left (\frac {1}{2} \sqrt {3} K[1]\right )\right )dK[1]+\left (3 \cos \left (\frac {\sqrt {3} t}{2}\right )-\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \int _1^t\frac {1}{3} e^{\frac {9 K[2]}{2}} \left (3 \cos \left (\frac {1}{2} \sqrt {3} K[2]\right ) (K[2]+1)-\sqrt {3} (K[2]-7) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )\right )dK[2]+3 c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )-2 \sqrt {3} c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )-\sqrt {3} c_2 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}
✓ Sympy. Time used: 0.435 (sec). Leaf size: 224
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(t + 4*x(t) - y(t) + Derivative(x(t), t) - 3,0),Eq(-t + x(t) + 5*y(t) + Derivative(y(t), t) - 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {4 t \sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} - \frac {4 t \cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} - \left (\frac {C_{1}}{2} - \frac {\sqrt {3} C_{2}}{2}\right ) e^{- \frac {9 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} + \left (\frac {\sqrt {3} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{- \frac {9 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + \frac {39 \sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{49} + \frac {39 \cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{49}, \ y{\left (t \right )} = C_{1} e^{- \frac {9 t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - C_{2} e^{- \frac {9 t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + \frac {5 t \sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} + \frac {5 t \cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{21} - \frac {\sin ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{147} - \frac {\cos ^{2}{\left (\frac {\sqrt {3} t}{2} \right )}}{147}\right ]
\]