50.29.11 problem 4(a)
Internal
problem
ID
[8207]
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(a)
Date
solved
:
Wednesday, March 05, 2025 at 05:32:14 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )-4 t +1\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+2 y \left (t \right )+3 t +4 \end{align*}
✓ Maple. Time used: 0.040 (sec). Leaf size: 105
ode:=[diff(x(t),t) = x(t)+2*y(t)-4*t+1, diff(y(t),t) = -x(t)+2*y(t)+3*t+4];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2} +{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1} +\frac {7 t}{2}+\frac {25}{8} \\
y &= \frac {{\mathrm e}^{\frac {3 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{4}+\frac {{\mathrm e}^{\frac {3 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}-\frac {{\mathrm e}^{\frac {3 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1}}{4}-\frac {5}{16}+\frac {t}{4} \\
\end{align*}
✓ Mathematica. Time used: 1.657 (sec). Leaf size: 128
ode={D[x[t],t]==x[t]+2*y[t]-4+t+1,D[y[t],t]==-x[t]+2*y[t]+3*t+4};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to t+c_1 e^{3 t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )-\frac {(c_1-4 c_2) e^{3 t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )}{\sqrt {7}}+\frac {9}{2} \\
y(t)\to -t+c_2 e^{3 t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )-\frac {(2 c_1-c_2) e^{3 t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )}{\sqrt {7}}-\frac {1}{4} \\
\end{align*}
✓ Sympy. Time used: 0.716 (sec). Leaf size: 224
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(4*t - x(t) - 2*y(t) + Derivative(x(t), t) - 1,0),Eq(-3*t + x(t) - 2*y(t) + Derivative(y(t), t) - 4,0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \frac {7 t \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} + \frac {7 t \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} - \left (\frac {C_{1}}{2} - \frac {\sqrt {7} C_{2}}{2}\right ) e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + \left (\frac {\sqrt {7} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {25 \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{8} + \frac {25 \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{8}, \ y{\left (t \right )} = - C_{1} e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + C_{2} e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {t \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4} + \frac {t \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4} - \frac {5 \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{16} - \frac {5 \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{16}\right ]
\]