87.20.8 problem 8

Internal problem ID [23693]
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 : 8
Date solved : Thursday, October 02, 2025 at 09:44:19 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 64

ode:=[diff(x__1(t),t) = x__1(t)+2*x__2(t), diff(x__2(t),t) = 4*x__1(t)+x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 113

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

\begin{align*} \text {x1}(t)&\to \frac {1}{4} e^{t-2 \sqrt {2} t} \left (2 c_1 \left (e^{4 \sqrt {2} t}+1\right )+\sqrt {2} c_2 \left (e^{4 \sqrt {2} t}-1\right )\right )\\ \text {x2}(t)&\to \frac {1}{2} e^{t-2 \sqrt {2} t} \left (\sqrt {2} c_1 \left (e^{4 \sqrt {2} t}-1\right )+c_2 \left (e^{4 \sqrt {2} t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 71

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

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