76.26.9 problem 9

Internal problem ID [17770]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.4 (Nondefective Matrices with Complex Eigenvalues). Problems at page 419
Problem number : 9
Date solved : Monday, March 31, 2025 at 04:27:11 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}+\frac {29 x_{2} \left (t \right )}{4}-\frac {11 x_{3} \left (t \right )}{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {3 x_{1} \left (t \right )}{4}+\frac {3 x_{2} \left (t \right )}{4}-\frac {5 x_{3} \left (t \right )}{2}\\ \frac {d}{d t}x_{3} \left (t \right )&=\frac {5 x_{1} \left (t \right )}{4}+\frac {11 x_{2} \left (t \right )}{4}-\frac {5 x_{3} \left (t \right )}{2} \end{align*}

✓ Maple. Time used: 0.151 (sec). Leaf size: 101

ode:=[diff(x__1(t),t) = 3/4*x__1(t)+29/4*x__2(t)-11/2*x__3(t), diff(x__2(t),t) = -3/4*x__1(t)+3/4*x__2(t)-5/2*x__3(t), diff(x__3(t),t) = 5/4*x__1(t)+11/4*x__2(t)-5/2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \sin \left (4 t \right )+c_3 \cos \left (4 t \right ) \\ x_{2} \left (t \right ) &= -c_1 \,{\mathrm e}^{-t}+\frac {c_2 \sin \left (4 t \right )}{5}+\frac {c_3 \cos \left (4 t \right )}{5}+\frac {2 c_2 \cos \left (4 t \right )}{5}-\frac {2 c_3 \sin \left (4 t \right )}{5} \\ x_{3} \left (t \right ) &= -c_1 \,{\mathrm e}^{-t}-\frac {c_2 \cos \left (4 t \right )}{5}+\frac {c_3 \sin \left (4 t \right )}{5}+\frac {2 c_2 \sin \left (4 t \right )}{5}+\frac {2 c_3 \cos \left (4 t \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 186

ode={D[x1[t],t]==3/4*x1[t]+29/4*x2[t]-11/2*x3[t],D[x2[t],t]==-3/4*x1[t]+3/4*x2[t]-5/2*x3[t],D[x3[t],t]==5/4*x1[t]+11/4*x2[t]-5/2*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left ((3 c_1+c_2+2 c_3) e^t \cos (4 t)+(c_1+7 c_2-6 c_3) e^t \sin (4 t)+c_1-c_2-2 c_3\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-t} \left ((c_1+3 c_2-2 c_3) e^t \cos (4 t)-(c_1-c_2+2 c_3) e^t \sin (4 t)-c_1+c_2+2 c_3\right ) \\ \text {x3}(t)\to \frac {1}{4} e^{-t} \left ((c_1-c_2+2 c_3) e^t \cos (4 t)+(c_1+3 c_2-2 c_3) e^t \sin (4 t)-c_1+c_2+2 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.169 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t)/4 - 29*x__2(t)/4 + 11*x__3(t)/2 + Derivative(x__1(t), t),0),Eq(3*x__1(t)/4 - 3*x__2(t)/4 + 5*x__3(t)/2 + Derivative(x__2(t), t),0),Eq(-5*x__1(t)/4 - 11*x__2(t)/4 + 5*x__3(t)/2 + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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