9.4.9 problem problem 9

Internal problem ID [973]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 9
Date solved : Tuesday, March 04, 2025 at 12:06:38 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

ode:=[diff(x__1(t),t) = 2*x__1(t)-5*x__2(t), diff(x__2(t),t) = 4*x__1(t)-2*x__2(t)]; 
ic:=x__1(0) = 2x__2(0) = 3; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {11 \sin \left (4 t \right )}{4}+2 \cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= 3 \cos \left (4 t \right )+\frac {\sin \left (4 t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 34

ode={D[ x1[t],t]==x1[t]-5*x2[t],D[ x2[t],t]==x1[t]-x2[t]}; 
ic={x1[0]==2,x2[0]==3}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 2 \cos (2 t)-13 \sin (t) \cos (t) \\ \text {x2}(t)\to 3 \cos (2 t)-\sin (t) \cos (t) \\ \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-2*x__1(t) + 5*x__2(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \left (\frac {C_{1}}{2} - C_{2}\right ) \cos {\left (4 t \right )} - \left (C_{1} + \frac {C_{2}}{2}\right ) \sin {\left (4 t \right )}, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (4 t \right )} - C_{2} \sin {\left (4 t \right )}\right ] \]