20.20.35 problem 39

Internal problem ID [3925]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 39
Date solved : Tuesday, March 04, 2025 at 05:19:33 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-7 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 47

ode:=[diff(x__1(t),t) = -3*x__1(t)-5*x__2(t), diff(x__2(t),t) = x__1(t)-7*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 54

ode={D[x1[t],t]==-3*x1[t]-5*x2[t],D[x2[t],t]==1*x1[t]-7*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-5 t} (c_1 \cos (t)+(2 c_1-5 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-5 t} (c_2 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(3*x__1(t) + 5*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 7*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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