10.16.13 problem 23

Internal problem ID [1413]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 23
Date solved : Saturday, March 29, 2025 at 10:54:35 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.158 (sec). Leaf size: 46

ode:=[diff(x__1(t),t) = -1/4*x__1(t)+x__2(t), diff(x__2(t),t) = -x__1(t)-1/4*x__2(t), diff(x__3(t),t) = -1/4*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (\sin \left (t \right ) c_1 +\cos \left (t \right ) c_2 \right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-\frac {t}{4}} \left (\sin \left (t \right ) c_2 -\cos \left (t \right ) c_1 \right ) \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-\frac {t}{4}} \\ \end{align*}

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 110

ode={D[ x1[t],t]==-1/4*x1[t]+1*x2[t]+0*x3[t],D[ x2[t],t]==-1*x1[t]-1/4*x2[t]+0*x3[t],D[ x3[t],t]==0*x1[t]-0*x2[t]-1/4*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+c_2 \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)-c_1 \sin (t)) \\ \text {x3}(t)\to c_3 e^{-t/4} \\ \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+c_2 \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)-c_1 \sin (t)) \\ \text {x3}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(x__1(t)/4 - x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) + x__2(t)/4 + Derivative(x__2(t), t),0),Eq(x__3(t)/4 + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- \frac {t}{4}} \sin {\left (t \right )} + C_{2} e^{- \frac {t}{4}} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = C_{1} e^{- \frac {t}{4}} \cos {\left (t \right )} - C_{2} e^{- \frac {t}{4}} \sin {\left (t \right )}, \ x^{3}{\left (t \right )} = C_{3} e^{- \frac {t}{4}}\right ] \]