87.20.25 problem 26

Internal problem ID [23710]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 161
Problem number : 26
Date solved : Thursday, October 02, 2025 at 09:44:29 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=2 x-y \left (t \right )\\ y^{\prime }\left (t \right )&=9 x+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 45

ode:=[diff(x(t),t) = 2*x(t)-y(t), diff(y(t),t) = 9*x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (c_2 \cos \left (3 t \right )+c_1 \sin \left (3 t \right )\right ) \\ y \left (t \right ) &= -3 \,{\mathrm e}^{2 t} \left (\cos \left (3 t \right ) c_1 -\sin \left (3 t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 56

ode={D[x[t],t]==2*x[t]-y[t],D[y[t],t]==9*x[t]+2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{3} e^{2 t} (3 c_1 \cos (3 t)-c_2 \sin (3 t))\\ y(t)&\to e^{2 t} (c_2 \cos (3 t)+3 c_1 \sin (3 t)) \end{align*}

✓ Sympy. Time used: 0.073 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + y(t) + Derivative(x(t), t),0),Eq(-9*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} e^{2 t} \sin {\left (3 t \right )}}{3} - \frac {C_{2} e^{2 t} \cos {\left (3 t \right )}}{3}, \ y{\left (t \right )} = C_{1} e^{2 t} \cos {\left (3 t \right )} - C_{2} e^{2 t} \sin {\left (3 t \right )}\right ] \]