57.19.8 problem 3(b)
Internal
problem
ID
[14503]
Book
:
A
First
Course
in
Differential
Equations
by
J.
David
Logan.
Third
Edition.
Springer-Verlag,
NY.
2015.
Section
:
Chapter
4,
Linear
Systems.
Exercises
page
202
Problem
number
:
3(b)
Date
solved
:
Thursday, October 02, 2025 at 09:37:56 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }&=3 y \left (t \right )-3 x\\ y^{\prime }\left (t \right )&=x+2 y \left (t \right )-1 \end{align*}
✓ Maple. Time used: 0.121 (sec). Leaf size: 87
ode:=[diff(x(t),t) = -3*x(t)+3*y(t), diff(y(t),t) = x(t)+2*y(t)-1];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_2 +{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_1 +\frac {1}{3} \\
y \left (t \right ) &= \frac {{\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_2 \sqrt {37}}{6}-\frac {{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_1 \sqrt {37}}{6}+\frac {5 \,{\mathrm e}^{\frac {\left (-1+\sqrt {37}\right ) t}{2}} c_2}{6}+\frac {5 \,{\mathrm e}^{-\frac {\left (1+\sqrt {37}\right ) t}{2}} c_1}{6}+\frac {1}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.261 (sec). Leaf size: 192
ode={D[x[t],t]==-3*x[t]+3*y[t],D[y[t],t]==x[t]+2*y[t]-1};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to \frac {1}{222} e^{-\frac {1}{2} \left (1+\sqrt {37}\right ) t} \left (74 e^{\frac {1}{2} \left (1+\sqrt {37}\right ) t}-3 \left (\left (5 \sqrt {37}-37\right ) c_1-6 \sqrt {37} c_2\right ) e^{\sqrt {37} t}+3 \left (\left (37+5 \sqrt {37}\right ) c_1-6 \sqrt {37} c_2\right )\right )\\ y(t)&\to \frac {1}{222} e^{-\frac {1}{2} \left (1+\sqrt {37}\right ) t} \left (74 e^{\frac {1}{2} \left (1+\sqrt {37}\right ) t}+3 \left (2 \sqrt {37} c_1+\left (37+5 \sqrt {37}\right ) c_2\right ) e^{\sqrt {37} t}-3 \left (2 \sqrt {37} c_1+\left (5 \sqrt {37}-37\right ) c_2\right )\right ) \end{align*}
✓ Sympy. Time used: 0.250 (sec). Leaf size: 82
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(3*x(t) - 3*y(t) + Derivative(x(t), t),0),Eq(-x(t) - 2*y(t) + Derivative(y(t), t) + 1,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (5 - \sqrt {37}\right ) e^{- \frac {t \left (1 - \sqrt {37}\right )}{2}}}{2} - \frac {C_{2} \left (5 + \sqrt {37}\right ) e^{- \frac {t \left (1 + \sqrt {37}\right )}{2}}}{2} + \frac {1}{3}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (1 - \sqrt {37}\right )}{2}} + C_{2} e^{- \frac {t \left (1 + \sqrt {37}\right )}{2}} + \frac {1}{3}\right ]
\]