9.6.3 problem problem 3

Internal problem ID [1010]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 3
Date solved : Saturday, March 29, 2025 at 10:36:48 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+5 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.100 (sec). Leaf size: 32

ode:=[diff(x__1(t),t) = x__1(t)-2*x__2(t), diff(x__2(t),t) = 2*x__1(t)+5*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 46

ode={D[ x1[t],t]==1*x1[t]-2*x2[t],D[ x2[t],t]==2*x1[t]+5*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.110 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 5*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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