87.19.1 problem 1

Internal problem ID [23671]
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 149
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:44:07 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+6 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 34

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__2(t), diff(x__2(t),t) = -3*x__1(t)+6*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{5 t} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+3 c_2 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 72

ode={D[x1[t],t]==2*x1[t]+x2[t],D[x2[t],t]==-3*x1[t]+6*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{2} e^{3 t} \left (c_2 \left (e^{2 t}-1\right )-c_1 \left (e^{2 t}-3\right )\right )\\ \text {x2}(t)&\to -\frac {1}{2} e^{3 t} \left (3 c_1 \left (e^{2 t}-1\right )+c_2 \left (1-3 e^{2 t}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-2*x1(t) - x2(t) + Derivative(x1(t), t),0),Eq(3*x1(t) - 6*x2(t) + Derivative(x2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = C_{1} e^{3 t} + \frac {C_{2} e^{5 t}}{3}, \ x_{2}{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{5 t}\right ] \]