63.23.4 problem 6
Internal
problem
ID
[13173]
Book
:
A
First
Course
in
Differential
Equations
by
J.
David
Logan.
Third
Edition.
Springer-Verlag,
NY.
2015.
Section
:
Chapter
4,
Linear
Systems.
Exercises
page
244
Problem
number
:
6
Date
solved
:
Monday, March 31, 2025 at 07:36:04 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+2 y \left (t \right )+3\\ \frac {d}{d t}y \left (t \right )&=7 x \left (t \right )+5 y \left (t \right )+2 t \end{align*}
✓ Maple. Time used: 0.129 (sec). Leaf size: 90
ode:=[diff(x(t),t) = 3*x(t)+2*y(t)+3, diff(y(t),t) = 7*x(t)+5*y(t)+2*t];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_2 +{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_1 +4 t +17 \\
y \left (t \right ) &= \frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_2 \sqrt {15}}{2}-\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_1 \sqrt {15}}{2}+\frac {{\mathrm e}^{\left (4+\sqrt {15}\right ) t} c_2}{2}+\frac {{\mathrm e}^{-\left (-4+\sqrt {15}\right ) t} c_1}{2}-6 t -25 \\
\end{align*}
✓ Mathematica. Time used: 1.702 (sec). Leaf size: 468
ode={D[x[t],t]==3*x[t]+2*y[t],D[y[t],t]==7*x[t]+5*y[t]+2*t};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{30} e^{-2 \sqrt {15} t} \left (-2 \sqrt {15} \left (e^{\left (4+\sqrt {15}\right ) t}-e^{\left (4+3 \sqrt {15}\right ) t}\right ) \int _1^t-\frac {1}{15} e^{-\left (\left (4+\sqrt {15}\right ) K[1]\right )} \left (-15-\sqrt {15}+\left (-15+\sqrt {15}\right ) e^{2 \sqrt {15} K[1]}\right ) K[1]dK[1]-6 \sqrt {15} t+22 t+e^{2 \sqrt {15} t} (76 t+604)+2 e^{4 \sqrt {15} t} \left (\left (11+3 \sqrt {15}\right ) t+23 \sqrt {15}+89\right )+\left (\left (15+\sqrt {15}\right ) c_1-2 \sqrt {15} c_2\right ) e^{\left (4+\sqrt {15}\right ) t}+\left (2 \sqrt {15} c_2-\left (\sqrt {15}-15\right ) c_1\right ) e^{\left (4+3 \sqrt {15}\right ) t}-46 \sqrt {15}+178\right ) \\
y(t)\to \frac {1}{30} e^{-\left (\left (\sqrt {15}-4\right ) t\right )} \left (\left (\left (15+\sqrt {15}\right ) e^{2 \sqrt {15} t}+15-\sqrt {15}\right ) \int _1^t-\frac {1}{15} e^{-\left (\left (4+\sqrt {15}\right ) K[1]\right )} \left (-15-\sqrt {15}+\left (-15+\sqrt {15}\right ) e^{2 \sqrt {15} K[1]}\right ) K[1]dK[1]+14 e^{-\left (\left (4+\sqrt {15}\right ) t\right )} \left (e^{2 \sqrt {15} t}-1\right ) \left (\left (\sqrt {15}-4\right ) t+e^{2 \sqrt {15} t} \left (\left (4+\sqrt {15}\right ) t+8 \sqrt {15}+31\right )+8 \sqrt {15}-31\right )+7 \sqrt {15} c_1 \left (e^{2 \sqrt {15} t}-1\right )+c_2 \left (\left (15+\sqrt {15}\right ) e^{2 \sqrt {15} t}+15-\sqrt {15}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.599 (sec). Leaf size: 78
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*x(t) - 2*y(t) + Derivative(x(t), t) - 3,0),Eq(-2*t - 7*x(t) - 5*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \frac {C_{1} \left (1 + \sqrt {15}\right ) e^{t \left (4 - \sqrt {15}\right )}}{7} - \frac {C_{2} \left (1 - \sqrt {15}\right ) e^{t \left (\sqrt {15} + 4\right )}}{7} + 4 t + 17, \ y{\left (t \right )} = C_{1} e^{t \left (4 - \sqrt {15}\right )} + C_{2} e^{t \left (\sqrt {15} + 4\right )} - 6 t - 25\right ]
\]