84.42.1 problem 31.8

Internal problem ID [22389]
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 : 31.8
Date solved : Thursday, October 02, 2025 at 08:38:09 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 ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 67

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = 6*x__1(t)]; 
ic:=[x__1(1) = 1, x__2(1) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 122

ode={D[x1[t],t]==x2[t],D[x2[t],t]==-2*x1[t]+8*x2[t]}; 
ic={x1[1]==1,x2[1]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.210 (sec). Leaf size: 119

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),0)] 
ics = {x1(1): 1, x2(1): 0} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

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