10.16.4 problem 4

Internal problem ID [1404]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 4
Date solved : Saturday, March 29, 2025 at 10:54:20 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {9 x_{1} \left (t \right )}{5}-x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.135 (sec). Leaf size: 57

ode:=[diff(x__1(t),t) = 2*x__1(t)-5/2*x__2(t), diff(x__2(t),t) = 9/5*x__1(t)-x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_1 +\cos \left (\frac {3 t}{2}\right ) c_2 \right ) \\ x_{2} \left (t \right ) &= \frac {3 \,{\mathrm e}^{\frac {t}{2}} \left (\sin \left (\frac {3 t}{2}\right ) c_1 +\sin \left (\frac {3 t}{2}\right ) c_2 -\cos \left (\frac {3 t}{2}\right ) c_1 +\cos \left (\frac {3 t}{2}\right ) c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 84

ode={D[ x1[t],t]==2*x1[t]-5/2*x2[t],D[ x2[t],t]==9/5*x1[t]-1*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{t/2} \left (3 c_1 \cos \left (\frac {3 t}{2}\right )+(3 c_1-5 c_2) \sin \left (\frac {3 t}{2}\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{t/2} \left (5 c_2 \cos \left (\frac {3 t}{2}\right )+(6 c_1-5 c_2) \sin \left (\frac {3 t}{2}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 75

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)/2 + Derivative(x__1(t), t),0),Eq(-9*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)
 

\[ \left [ x^{1}{\left (t \right )} = \left (\frac {5 C_{1}}{6} - \frac {5 C_{2}}{6}\right ) e^{\frac {t}{2}} \cos {\left (\frac {3 t}{2} \right )} - \left (\frac {5 C_{1}}{6} + \frac {5 C_{2}}{6}\right ) e^{\frac {t}{2}} \sin {\left (\frac {3 t}{2} \right )}, \ x^{2}{\left (t \right )} = C_{1} e^{\frac {t}{2}} \cos {\left (\frac {3 t}{2} \right )} - C_{2} e^{\frac {t}{2}} \sin {\left (\frac {3 t}{2} \right )}\right ] \]