87.20.10 problem 11

Internal problem ID [23695]
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 : 11
Date solved : Thursday, October 02, 2025 at 09:44:21 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=1 \\ x_{2} \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = 4*x__1(t)-x__2(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)]; 
ic:=[x__1(0) = 1, x__2(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{2 t}-{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= 4 \,{\mathrm e}^{2 t}-{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 32

ode={D[x1[t],t]==4*x1[t]-x2[t],D[x2[t],t]==2*x1[t]+x2[t]}; 
ic={x1[0]==1,x2[0]==3}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to -e^{2 t} \left (e^t-2\right )\\ \text {x2}(t)&\to -e^{2 t} \left (e^t-4\right ) \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-4*x1(t) + x2(t) + Derivative(x1(t), t),0),Eq(-2*x1(t) - x2(t) + Derivative(x2(t), t),0)] 
ics = {x1(0): 1, x2(0): 3} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = - e^{3 t} + 2 e^{2 t}, \ x_{2}{\left (t \right )} = - e^{3 t} + 4 e^{2 t}\right ] \]