4.18.10 problem 10

Internal problem ID [1437]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 04:34:05 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.246 (sec). Leaf size: 89

ode:=[diff(x__1(t),t) = -3*x__1(t)+2^(1/2)*x__2(t)+exp(-t), diff(x__2(t),t) = 2^(1/2)*x__1(t)-2*x__2(t)-exp(-t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-4 t} c_2 +{\mathrm e}^{-t} c_1 -\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}+\frac {t \,{\mathrm e}^{-t}}{3} \\ x_{2} \left (t \right ) &= -\frac {2 t \,{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-t}}{3}+\sqrt {2}\, {\mathrm e}^{-t} c_1 +\frac {t \,{\mathrm e}^{-t} \sqrt {2}}{3}-\frac {\sqrt {2}\, {\mathrm e}^{-4 t} c_2}{2}-\frac {\sqrt {2}\, {\mathrm e}^{-t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 128

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

\begin{align*} \text {x1}(t)&\to \frac {1}{9} e^{-4 t} \left (e^{3 t} \left (-3 \left (\sqrt {2}-1\right ) t+\sqrt {2}+2+3 c_1+3 \sqrt {2} c_2\right )+6 c_1-3 \sqrt {2} c_2\right )\\ \text {x2}(t)&\to \frac {1}{9} e^{-4 t} \left (e^{3 t} \left (3 \left (\sqrt {2}-2\right ) t-\sqrt {2}-1+3 \sqrt {2} c_1+6 c_2\right )-3 \sqrt {2} c_1+3 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.190 (sec). Leaf size: 83

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

\[ \left [ x^{1}{\left (t \right )} = - \sqrt {2} C_{2} e^{- 4 t} + \frac {t \left (1 - \sqrt {2}\right ) e^{- t}}{3} + \left (\frac {\sqrt {2} C_{1}}{2} + \frac {\sqrt {2} + 2}{9}\right ) e^{- t}, \ x^{2}{\left (t \right )} = C_{2} e^{- 4 t} - \frac {t \left (2 - \sqrt {2}\right ) e^{- t}}{3} + \left (C_{1} - \frac {1 + \sqrt {2}}{9}\right ) e^{- t}\right ] \]