52.9.7 problem 7
Internal
problem
ID
[8385]
Book
:
DIFFERENTIAL
EQUATIONS
with
Boundary
Value
Problems.
DENNIS
G.
ZILL,
WARREN
S.
WRIGHT,
MICHAEL
R.
CULLEN.
Brooks/Cole.
Boston,
MA.
2013.
8th
edition.
Section
:
CHAPTER
8
SYSTEMS
OF
LINEAR
FIRST-ORDER
DIFFERENTIAL
EQUATIONS.
EXERCISES
8.1.
Page
332
Problem
number
:
7
Date
solved
:
Sunday, March 30, 2025 at 12:56:44 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+2 y \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+3 y \left (t \right )-{\mathrm e}^{t} \end{align*}
✓ Maple. Time used: 0.201 (sec). Leaf size: 105
ode:=[diff(x(t),t) = 4*x(t)+2*y(t)+exp(t), diff(y(t),t) = -x(t)+3*y(t)-exp(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {7 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_2 +{\mathrm e}^{\frac {7 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_1 -\frac {{\mathrm e}^{t}}{2} \\
y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {7 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_2}{4}+\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_2}{4}-\frac {{\mathrm e}^{\frac {7 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_1}{4}-\frac {{\mathrm e}^{\frac {7 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_1}{4}+\frac {{\mathrm e}^{t}}{4} \\
\end{align*}
✓ Mathematica. Time used: 0.397 (sec). Leaf size: 129
ode={D[x[t],t]==4*x[t]+2*y[t]+Exp[t],D[y[t],t]==-x[t]+3*y[t]-Exp[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to -\frac {e^t}{2}+c_1 e^{7 t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )+\frac {(c_1+4 c_2) e^{7 t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )}{\sqrt {7}} \\
y(t)\to \frac {e^t}{4}+c_2 e^{7 t/2} \cos \left (\frac {\sqrt {7} t}{2}\right )-\frac {(2 c_1+c_2) e^{7 t/2} \sin \left (\frac {\sqrt {7} t}{2}\right )}{\sqrt {7}} \\
\end{align*}
✓ Sympy. Time used: 0.501 (sec). Leaf size: 167
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-4*x(t) - 2*y(t) - exp(t) + Derivative(x(t), t),0),Eq(x(t) - 3*y(t) + exp(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} + \frac {\sqrt {7} C_{2}}{2}\right ) e^{\frac {7 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + \left (\frac {\sqrt {7} C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{\frac {7 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} - \frac {e^{t} \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} - \frac {e^{t} \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2}, \ y{\left (t \right )} = - C_{1} e^{\frac {7 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + C_{2} e^{\frac {7 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {e^{t} \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4} + \frac {e^{t} \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{4}\right ]
\]