44.29.12 problem 4(b)
Internal
problem
ID
[9494]
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(b)
Date
solved
:
Tuesday, September 30, 2025 at 06:19:36 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 x \left (t \right )+y \left (t \right )-t +3\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+4 y \left (t \right )+t -2 \end{align*}
✓ Maple. Time used: 0.130 (sec). Leaf size: 89
ode:=[diff(x(t),t) = -2*x(t)+y(t)-t+3, diff(y(t),t) = x(t)+4*y(t)+t-2];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_2 +{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_1 -\frac {5 t}{9}+\frac {145}{81} \\
y \left (t \right ) &= {\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_2 \sqrt {10}-{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_1 \sqrt {10}+3 \,{\mathrm e}^{\left (1+\sqrt {10}\right ) t} c_2 +3 \,{\mathrm e}^{-\left (-1+\sqrt {10}\right ) t} c_1 -\frac {t}{9}+\frac {2}{81} \\
\end{align*}
✓ Mathematica. Time used: 10.757 (sec). Leaf size: 556
ode={D[x[t],t]==-2*x[t]+y[t]-t+3,D[y[t],t]==x[t]+4*y[t]+t-2};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{20} e^{t-\sqrt {10} t} \left (-\left (\left (3 \sqrt {10}-10\right ) e^{2 \sqrt {10} t}-10-3 \sqrt {10}\right ) \int _1^t\frac {1}{20} e^{-\left (\left (1+\sqrt {10}\right ) K[1]\right )} \left (2 \left (-5+2 \sqrt {10}\right ) K[1]+e^{2 \sqrt {10} K[1]} \left (-2 \left (5+2 \sqrt {10}\right ) K[1]+11 \sqrt {10}+30\right )-11 \sqrt {10}+30\right )dK[1]+\sqrt {10} \left (e^{2 \sqrt {10} t}-1\right ) \int _1^t\frac {1}{20} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (5+\sqrt {10}\right ) K[2]+e^{2 \sqrt {10} K[2]} \left (-2 \left (-5+\sqrt {10}\right ) K[2]+3 \sqrt {10}-20\right )-3 \sqrt {10}-20\right )dK[2]-\left (c_1 \left (\left (3 \sqrt {10}-10\right ) e^{2 \sqrt {10} t}-10-3 \sqrt {10}\right )\right )+\sqrt {10} c_2 \left (e^{2 \sqrt {10} t}-1\right )\right )\\ y(t)&\to \frac {1}{20} e^{t-\sqrt {10} t} \left (\sqrt {10} \left (e^{2 \sqrt {10} t}-1\right ) \int _1^t\frac {1}{20} e^{-\left (\left (1+\sqrt {10}\right ) K[1]\right )} \left (2 \left (-5+2 \sqrt {10}\right ) K[1]+e^{2 \sqrt {10} K[1]} \left (-2 \left (5+2 \sqrt {10}\right ) K[1]+11 \sqrt {10}+30\right )-11 \sqrt {10}+30\right )dK[1]+\left (\left (10+3 \sqrt {10}\right ) e^{2 \sqrt {10} t}+10-3 \sqrt {10}\right ) \int _1^t\frac {1}{20} e^{-\left (\left (1+\sqrt {10}\right ) K[2]\right )} \left (2 \left (5+\sqrt {10}\right ) K[2]+e^{2 \sqrt {10} K[2]} \left (-2 \left (-5+\sqrt {10}\right ) K[2]+3 \sqrt {10}-20\right )-3 \sqrt {10}-20\right )dK[2]+\sqrt {10} c_1 \left (e^{2 \sqrt {10} t}-1\right )+c_2 \left (\left (10+3 \sqrt {10}\right ) e^{2 \sqrt {10} t}+10-3 \sqrt {10}\right )\right ) \end{align*}
✓ Sympy. Time used: 0.380 (sec). Leaf size: 80
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(t + 2*x(t) - y(t) + Derivative(x(t), t) - 3,0),Eq(-t - x(t) - 4*y(t) + Derivative(y(t), t) + 2,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - C_{1} \left (3 - \sqrt {10}\right ) e^{t \left (1 + \sqrt {10}\right )} - C_{2} \left (3 + \sqrt {10}\right ) e^{t \left (1 - \sqrt {10}\right )} - \frac {5 t}{9} + \frac {145}{81}, \ y{\left (t \right )} = C_{1} e^{t \left (1 + \sqrt {10}\right )} + C_{2} e^{t \left (1 - \sqrt {10}\right )} - \frac {t}{9} + \frac {2}{81}\right ]
\]