20.18.6 problem 6

Internal problem ID [3876]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.6 (Variation of parameters for linear systems), page 624
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 05:18:29 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )+4 x_{2} \left (t \right )+8 \sin \left (2 t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-2 x_{2} \left (t \right )+8 \cos \left (2 t \right ) \end{align*}

✓ Maple. Time used: 2.692 (sec). Leaf size: 95

ode:=[diff(x__1(t),t) = 2*x__1(t)+4*x__2(t)+8*sin(2*t), diff(x__2(t),t) = -2*x__1(t)-2*x__2(t)+8*cos(2*t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_{1} \cos \left (2 t \right )-4 t \cos \left (2 t \right )+c_{2} \sin \left (2 t \right )+12 t \sin \left (2 t \right )+6 \cos \left (2 t \right ) \\ x_{2} \left (t \right ) &= -\frac {c_{1} \cos \left (2 t \right )}{2}+\frac {c_{2} \cos \left (2 t \right )}{2}+8 t \cos \left (2 t \right )-\frac {c_{1} \sin \left (2 t \right )}{2}-\frac {c_{2} \sin \left (2 t \right )}{2}-4 t \sin \left (2 t \right )-4 \cos \left (2 t \right )-2 \sin \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 64

ode={D[x1[t],t]==2*x1[t]+4*x2[t]+8*Sin[2*t],D[x2[t],t]==-2*x1[t]-2*x2[t]+8*Cos[2*t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to (-4 t+1+c_1) \cos (2 t)+(12 t+1+c_1+2 c_2) \sin (2 t) \\ \text {x2}(t)\to (8 t-1+c_2) \cos (2 t)-(4 t+c_1+c_2) \sin (2 t) \\ \end{align*}

✓ Sympy. Time used: 0.143 (sec). Leaf size: 90

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-2*x__1(t) - 4*x__2(t) - 8*sin(2*t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) + 2*x__2(t) - 8*cos(2*t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 12 t \sin {\left (2 t \right )} - 4 t \cos {\left (2 t \right )} - \left (C_{1} - C_{2}\right ) \cos {\left (2 t \right )} + \left (C_{1} + C_{2}\right ) \sin {\left (2 t \right )} + 2 \sin ^{3}{\left (2 t \right )} + 2 \sin {\left (2 t \right )} \cos ^{2}{\left (2 t \right )}, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (2 t \right )} - C_{2} \sin {\left (2 t \right )} - 4 t \sin {\left (2 t \right )} + 8 t \cos {\left (2 t \right )}\right ] \]