72.11.12 problem 14
Internal
problem
ID
[14766]
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
:
14
Date
solved
:
Thursday, March 13, 2025 at 04:18:34 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-3 x \left (t \right )+2 y \left (t \right ) \end{align*}
With initial conditions
\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -1 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 60
ode:=[diff(x(t),t) = x(t)+4*y(t), diff(y(t),t) = -3*x(t)+2*y(t)];
ic:=x(0) = 1y(0) = -1;
dsolve([ode,ic]);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (-\frac {9 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+\cos \left (\frac {\sqrt {47}\, t}{2}\right )\right ) \\
y &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\frac {56 \sqrt {47}\, \sin \left (\frac {\sqrt {47}\, t}{2}\right )}{47}+8 \cos \left (\frac {\sqrt {47}\, t}{2}\right )\right )}{8} \\
\end{align*}
✓ Mathematica. Time used: 0.016 (sec). Leaf size: 94
ode={D[x[t],t]==1*x[t]+4*y[t],D[y[t],t]==-3*x[t]+2*y[t]};
ic={x[0]==1,y[0]==-1};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {1}{47} e^{3 t/2} \left (47 \cos \left (\frac {\sqrt {47} t}{2}\right )-9 \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: 0.192 (sec). Leaf size: 99
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-x(t) - 4*y(t) + Derivative(x(t), t),0),Eq(3*x(t) - 2*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = \left (\frac {C_{1}}{6} + \frac {\sqrt {47} C_{2}}{6}\right ) e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {47} t}{2} \right )} + \left (\frac {\sqrt {47} C_{1}}{6} - \frac {C_{2}}{6}\right ) e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {47} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {3 t}{2}} \cos {\left (\frac {\sqrt {47} t}{2} \right )} - C_{2} e^{\frac {3 t}{2}} \sin {\left (\frac {\sqrt {47} t}{2} \right )}\right ]
\]