20.18.4 problem 4

Internal problem ID [3874]
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 : 4
Date solved : Tuesday, March 04, 2025 at 05:18:27 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+x_{2} \left (t \right )+20 \,{\mathrm e}^{3 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )+x_{2} \left (t \right )+12 \,{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 55

ode:=[diff(x__1(t),t) = -x__1(t)+x__2(t)+20*exp(3*t), diff(x__2(t),t) = 3*x__1(t)+x__2(t)+12*exp(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-2 t} c_{2} +{\mathrm e}^{2 t} c_{1} +8 \,{\mathrm e}^{3 t}-4 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= 12 \,{\mathrm e}^{3 t}-{\mathrm e}^{-2 t} c_{2} +3 \,{\mathrm e}^{2 t} c_{1} -8 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.819 (sec). Leaf size: 135

ode={D[x1[t],t]==x1[t]+x2[t]+20*Exp[3*t],D[x2[t],t]==3*x1[t]+x2[t]+12*Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{6} e^t \left (240 e^{2 t}+\left (3 c_1-\sqrt {3} c_2\right ) e^{-\sqrt {3} t}+\left (3 c_1+\sqrt {3} c_2\right ) e^{\sqrt {3} t}-24\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{t-\sqrt {3} t} \left (120 e^{\left (2+\sqrt {3}\right ) t}+\left (\sqrt {3} c_1+c_2\right ) e^{2 \sqrt {3} t}-\sqrt {3} c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.200 (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) - x__2(t) - 20*exp(3*t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) - x__2(t) - 12*exp(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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