72.11.5 problem 7
Internal
problem
ID
[14759]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
3.
Linear
Systems.
Exercises
section
3.4
page
310
Problem
number
:
7
Date
solved
:
Thursday, March 13, 2025 at 04:18:24 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-6 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+y \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 1 \end{align*}
✓ Maple. Time used: 0.069 (sec). Leaf size: 62
ode:=[diff(x(t),t) = 2*x(t)-6*y(t), diff(y(t),t) = 2*x(t)+y(t)];
ic:=x(0) = 2y(0) = 1;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {10 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+2 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\
y &= \frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {84 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+12 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{12} \\
\end{align*}
✓ Mathematica. Time used: 0.016 (sec). Leaf size: 94
ode={D[x[t],t]==2*x[t]-6*y[t],D[y[t],t]==2*x[t]+y[t]};
ic={x[0]==2,y[0]==1};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {2}{47} e^{3 t/2} \left (47 \cos \left (\frac {\sqrt {47} t}{2}\right )-5 \sqrt {47} \sin \left (\frac {\sqrt {47} t}{2}\right )\right ) \\
y(t)\to \frac {1}{47} e^{3 t/2} \left (7 \sqrt {47} \sin \left (\frac {\sqrt {47} t}{2}\right )+47 \cos \left (\frac {\sqrt {47} t}{2}\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 0.203 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*x(t) + 6*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) - y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{4} - \frac {\sqrt {47} C_{2}}{4}\right ) e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {47} t}{2} \right )} - \left (\frac {\sqrt {47} C_{1}}{4} + \frac {C_{2}}{4}\right ) e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {47} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {47} t}{2} \right )} - C_{2} e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {47} t}{2} \right )}\right ]
\]