20.18.5 problem 5

Internal problem ID [3875]
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 : 5
Date solved : Sunday, March 30, 2025 at 02:10:37 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )+54 t \,{\mathrm e}^{3 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+4 x_{2} \left (t \right )+9 \,{\mathrm e}^{3 t} \end{align*}

✓ Maple. Time used: 0.204 (sec). Leaf size: 60

ode:=[diff(x__1(t),t) = -x__1(t)+2*x__2(t)+54*t*exp(3*t), diff(x__2(t),t) = -2*x__1(t)+4*x__2(t)+9*exp(3*t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \left (-9 t^{2}+\frac {1}{3} c_1 +30 t -10\right ) {\mathrm e}^{3 t}+c_2 \\ x_{2} \left (t \right ) &= 24 t \,{\mathrm e}^{3 t}-18 \,{\mathrm e}^{3 t} t^{2}+\frac {2 \,{\mathrm e}^{3 t} c_1}{3}-5 \,{\mathrm e}^{3 t}+\frac {c_2}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 81

ode={D[x1[t],t]==-x1[t]+2*x2[t]+54*t*Exp[3*t],D[x2[t],t]==-2*x1[t]+4*x2[t]+9*Exp[3*t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} \left (-e^{3 t} \left (27 t^2-90 t+30+c_1-2 c_2\right )+4 c_1-2 c_2\right ) \\ \text {x2}(t)\to \frac {1}{3} \left (e^{3 t} \left (-54 t^2+72 t-15-2 c_1+4 c_2\right )+2 c_1-c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-54*t*exp(3*t) + x__1(t) - 2*x__2(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) - 4*x__2(t) - 9*exp(3*t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = 2 C_{1} - 9 t^{2} e^{3 t} + 30 t e^{3 t} + \left (\frac {C_{2}}{2} - 10\right ) e^{3 t}, \ x^{2}{\left (t \right )} = C_{1} - 18 t^{2} e^{3 t} + 24 t e^{3 t} + \left (C_{2} - 5\right ) e^{3 t}\right ] \]