84.42.5 problem 31.12

Internal problem ID [22393]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 31. Solutions of linear systems with constant coefficients. Supplementary problems
Problem number : 31.12
Date solved : Thursday, October 02, 2025 at 08:38:12 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=6 x_{1} \left (t \right )+9 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=0 \\ x_{2} \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 86

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = 6*x__1(t)+9*exp(-t)]; 
ic:=[x__1(0) = 0, x__2(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\sqrt {6}\, t} \left (\frac {9}{10}-\frac {3 \sqrt {6}}{20}\right )+{\mathrm e}^{-\sqrt {6}\, t} \left (\frac {9}{10}+\frac {3 \sqrt {6}}{20}\right )-\frac {9 \,{\mathrm e}^{-t}}{5} \\ x_{2} \left (t \right ) &= \sqrt {6}\, {\mathrm e}^{\sqrt {6}\, t} \left (\frac {9}{10}-\frac {3 \sqrt {6}}{20}\right )-\sqrt {6}\, {\mathrm e}^{-\sqrt {6}\, t} \left (\frac {9}{10}+\frac {3 \sqrt {6}}{20}\right )+\frac {9 \,{\mathrm e}^{-t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.341 (sec). Leaf size: 124

ode={D[x1[t],t]==x2[t],D[x2[t],t]==-2*x1[t]+8*x2[t]+9*Exp[-t]}; 
ic={x1[0]==0,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {9}{308} e^{-\left (\left (\sqrt {14}-4\right ) t\right )} \left (\left (5 \sqrt {14}-14\right ) e^{2 \sqrt {14} t}+28 e^{\left (\sqrt {14}-5\right ) t}-14-5 \sqrt {14}\right )\\ \text {x2}(t)&\to \frac {9}{154} e^{-\left (\left (\sqrt {14}-4\right ) t\right )} \left (\left (7+3 \sqrt {14}\right ) e^{2 \sqrt {14} t}-14 e^{\left (\sqrt {14}-5\right ) t}+7-3 \sqrt {14}\right ) \end{align*}

✓ Sympy. Time used: 0.809 (sec). Leaf size: 105

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-x2(t) + Derivative(x1(t), t),0),Eq(2*x1(t) - 8*x2(t) + Derivative(x2(t), t) - 9*exp(-t),0)] 
ics = {x1(0): 0, x2(0): 0} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = - \frac {9 \left (14 + 5 \sqrt {14}\right ) e^{t \left (4 - \sqrt {14}\right )}}{308} - \frac {9 \left (14 - 5 \sqrt {14}\right ) e^{t \left (\sqrt {14} + 4\right )}}{308} + \frac {9 e^{- t}}{11}, \ x_{2}{\left (t \right )} = \frac {9 \left (7 - 3 \sqrt {14}\right ) e^{t \left (4 - \sqrt {14}\right )}}{154} + \frac {9 \left (7 + 3 \sqrt {14}\right ) e^{t \left (\sqrt {14} + 4\right )}}{154} - \frac {9 e^{- t}}{11}\right ] \]