52.9.1 problem 1
Internal
problem
ID
[8379]
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
:
1
Date
solved
:
Sunday, March 30, 2025 at 12:53:25 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )+8 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.147 (sec). Leaf size: 83
ode:=[diff(x(t),t) = 3*x(t)-5*y(t), diff(y(t),t) = 4*x(t)+8*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {11 t}{2}} \left (\sin \left (\frac {\sqrt {55}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {55}\, t}{2}\right ) c_2 \right ) \\
y \left (t \right ) &= \frac {{\mathrm e}^{\frac {11 t}{2}} \left (\sin \left (\frac {\sqrt {55}\, t}{2}\right ) \sqrt {55}\, c_2 -\cos \left (\frac {\sqrt {55}\, t}{2}\right ) \sqrt {55}\, c_1 -5 \sin \left (\frac {\sqrt {55}\, t}{2}\right ) c_1 -5 \cos \left (\frac {\sqrt {55}\, t}{2}\right ) c_2 \right )}{10} \\
\end{align*}
✓ Mathematica. Time used: 0.026 (sec). Leaf size: 113
ode={D[x[t],t]==3*x[t]-5*y[t],D[y[t],t]==4*x[t]+8*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{11} e^{11 t/2} \left (11 c_1 \cos \left (\frac {\sqrt {55} t}{2}\right )-\sqrt {55} (c_1+2 c_2) \sin \left (\frac {\sqrt {55} t}{2}\right )\right ) \\
y(t)\to \frac {1}{55} e^{11 t/2} \left (55 c_2 \cos \left (\frac {\sqrt {55} t}{2}\right )+\sqrt {55} (8 c_1+5 c_2) \sin \left (\frac {\sqrt {55} t}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.206 (sec). Leaf size: 104
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-3*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) - 8*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {5 C_{1}}{8} + \frac {\sqrt {55} C_{2}}{8}\right ) e^{\frac {11 t}{2}} \cos {\left (\frac {\sqrt {55} t}{2} \right )} - \left (\frac {\sqrt {55} C_{1}}{8} - \frac {5 C_{2}}{8}\right ) e^{\frac {11 t}{2}} \sin {\left (\frac {\sqrt {55} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {11 t}{2}} \cos {\left (\frac {\sqrt {55} t}{2} \right )} - C_{2} e^{\frac {11 t}{2}} \sin {\left (\frac {\sqrt {55} t}{2} \right )}\right ]
\]