10.16.11 problem 11

Internal problem ID [1411]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 12:35:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=\frac {3 x_{1} \left (t \right )}{4}-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{4} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 47

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (c_1 \sin \left (t \right )+c_2 \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{4}} \left (\cos \left (t \right ) c_1 -c_2 \cos \left (t \right )-c_1 \sin \left (t \right )-\sin \left (t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 56

ode={D[ x1[t],t]==3/4*x1[t]-2*x2[t],D[ x2[t],t]==1*x1[t]-5/4*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)+(c_1-c_2) \sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t)/4 + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 5*x__2(t)/4 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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