10.16.15 problem 25

Internal problem ID [1415]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 25
Date solved : Saturday, March 29, 2025 at 10:54:37 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {x_{1} \left (t \right )}{2}-\frac {x_{2} \left (t \right )}{8}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2} \end{align*}

✓ Maple. Time used: 0.136 (sec). Leaf size: 45

ode:=[diff(x__1(t),t) = -1/2*x__1(t)-1/8*x__2(t), diff(x__2(t),t) = 2*x__1(t)-1/2*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 68

ode={D[ x1[t],t]==-1/2*x1[t]-1/8*x2[t],D[ x2[t],t]==2*x1[t]-1/2*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} \left (4 c_1 \cos \left (\frac {t}{2}\right )-c_2 \sin \left (\frac {t}{2}\right )\right ) \\ \text {x2}(t)\to e^{-t/2} \left (c_2 \cos \left (\frac {t}{2}\right )+4 c_1 \sin \left (\frac {t}{2}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 56

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)/8 + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + x__2(t)/2 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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