84.42.2 problem 31.9

Internal problem ID [22390]
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 : 31.9
Date solved : Thursday, October 02, 2025 at 08:38:10 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 )+4 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 48

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = 6*x__1(t)+4]; 
ic:=[x__1(0) = 0, x__2(0) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.346 (sec). Leaf size: 98

ode={D[x1[t],t]==x2[t],D[x2[t],t]==-2*x1[t]+8*x2[t]+4}; 
ic={x1[0]==0,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \frac {1}{7} e^{-\left (\left (\sqrt {14}-4\right ) t\right )} \left (\left (2 \sqrt {14}-7\right ) e^{2 \sqrt {14} t}+14 e^{\left (\sqrt {14}-4\right ) t}-7-2 \sqrt {14}\right )\\ \text {x2}(t)&\to \sqrt {\frac {2}{7}} e^{-\left (\left (\sqrt {14}-4\right ) t\right )} \left (e^{2 \sqrt {14} t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.516 (sec). Leaf size: 80

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) - 4,0)] 
ics = {x1(0): 0, x2(0): 0} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

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