69.1.132 problem 191
Internal
problem
ID
[14214]
Book
:
DIFFERENTIAL
and
INTEGRAL
CALCULUS.
VOL
I.
by
N.
PISKUNOV.
MIR
PUBLISHERS,
Moscow
1969.
Section
:
Chapter
8.
Differential
equations.
Exercises
page
595
Problem
number
:
191
Date
solved
:
Monday, March 31, 2025 at 12:12:54 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )+6 y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.137 (sec). Leaf size: 77
ode:=[diff(x(t),t) = 2*x(t)-3*y(t), diff(y(t),t) = 5*x(t)+6*y(t)];
dsolve(ode);
\begin{align*}
x \left (t \right ) &= {\mathrm e}^{4 t} \left (\sin \left (\sqrt {11}\, t \right ) c_1 +\cos \left (\sqrt {11}\, t \right ) c_2 \right ) \\
y \left (t \right ) &= \frac {{\mathrm e}^{4 t} \left (\sin \left (\sqrt {11}\, t \right ) \sqrt {11}\, c_2 -\cos \left (\sqrt {11}\, t \right ) \sqrt {11}\, c_1 -2 \sin \left (\sqrt {11}\, t \right ) c_1 -2 \cos \left (\sqrt {11}\, t \right ) c_2 \right )}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.015 (sec). Leaf size: 99
ode={D[x[t],t]==2*x[t]-3*y[t],D[y[t],t]==5*x[t]+6*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to c_1 e^{4 t} \cos \left (\sqrt {11} t\right )-\frac {(2 c_1+3 c_2) e^{4 t} \sin \left (\sqrt {11} t\right )}{\sqrt {11}} \\
y(t)\to c_2 e^{4 t} \cos \left (\sqrt {11} t\right )+\frac {(5 c_1+2 c_2) e^{4 t} \sin \left (\sqrt {11} t\right )}{\sqrt {11}} \\
\end{align*}
✓ Sympy. Time used: 0.200 (sec). Leaf size: 90
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-2*x(t) + 3*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) - 6*y(t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (\frac {2 C_{1}}{5} + \frac {\sqrt {11} C_{2}}{5}\right ) e^{4 t} \cos {\left (\sqrt {11} t \right )} - \left (\frac {\sqrt {11} C_{1}}{5} - \frac {2 C_{2}}{5}\right ) e^{4 t} \sin {\left (\sqrt {11} t \right )}, \ y{\left (t \right )} = C_{1} e^{4 t} \cos {\left (\sqrt {11} t \right )} - C_{2} e^{4 t} \sin {\left (\sqrt {11} t \right )}\right ]
\]