14.25.9 problem 10

Internal problem ID [2766]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : 10
Date solved : Sunday, March 30, 2025 at 12:16:51 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.178 (sec). Leaf size: 81

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\sqrt {7}\, t} c_2 +{\mathrm e}^{-\sqrt {7}\, t} c_1 +3 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= \sqrt {7}\, {\mathrm e}^{\sqrt {7}\, t} c_2 -\sqrt {7}\, {\mathrm e}^{-\sqrt {7}\, t} c_1 +2 \,{\mathrm e}^{3 t}-2 \,{\mathrm e}^{\sqrt {7}\, t} c_2 -2 \,{\mathrm e}^{-\sqrt {7}\, t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.599 (sec). Leaf size: 171

ode={D[ x1[t],t]==2*x1[t]+1*x2[t]+Exp[3*t],D[ x2[t],t]==3*x1[t]-2*x2[t]+Exp[3*t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{14} e^{-\sqrt {7} t} \left (42 e^{\left (3+\sqrt {7}\right ) t}+\left (\left (7+2 \sqrt {7}\right ) c_1+\sqrt {7} c_2\right ) e^{2 \sqrt {7} t}+\left (7-2 \sqrt {7}\right ) c_1-\sqrt {7} c_2\right ) \\ \text {x2}(t)\to \frac {1}{14} e^{-\sqrt {7} t} \left (28 e^{\left (3+\sqrt {7}\right ) t}+\left (3 \sqrt {7} c_1+\left (7-2 \sqrt {7}\right ) c_2\right ) e^{2 \sqrt {7} t}-3 \sqrt {7} c_1+\left (7+2 \sqrt {7}\right ) c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.521 (sec). Leaf size: 75

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

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} \left (2 + \sqrt {7}\right ) e^{\sqrt {7} t}}{3} + \frac {C_{2} \left (2 - \sqrt {7}\right ) e^{- \sqrt {7} t}}{3} + 3 e^{3 t}, \ x^{2}{\left (t \right )} = C_{1} e^{\sqrt {7} t} + C_{2} e^{- \sqrt {7} t} + 2 e^{3 t}\right ] \]