84.41.1 problem 31.1

Internal problem ID [22383]
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. Solved problems. Page 194
Problem number : 31.1
Date solved : Thursday, October 02, 2025 at 08:38:05 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=y\\ y^{\prime }&=8 x \left (t \right )-2 y \end{align*}

With initial conditions

\begin{align*} x \left (1\right )&=2 \\ y \left (1\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 49

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = 8*x(t)-2*y(t)]; 
ic:=[x(1) = 2, y(1) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{4} {\mathrm e}^{-4 t}}{6}+\frac {11 \,{\mathrm e}^{-2} {\mathrm e}^{2 t}}{6} \\ y \left (t \right ) &= -\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{-4 t}}{3}+\frac {11 \,{\mathrm e}^{-2} {\mathrm e}^{2 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 53

ode={D[x[t],t]==y[t],D[y[t],t]==8*x[t]-2*y[t]}; 
ic={x[1]==2,y[1]==3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} e^{-4 t-2} \left (11 e^{6 t}+e^6\right )\\ y(t)&\to \frac {11}{3} e^{2 t-2}-\frac {2}{3} e^{4-4 t} \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(-8*x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {x(1): 2, y(1): 3} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {11 e^{2 t}}{6 e^{2}} + \frac {e^{4} e^{- 4 t}}{6}, \ y{\left (t \right )} = \frac {11 e^{2 t}}{3 e^{2}} - \frac {2 e^{4} e^{- 4 t}}{3}\right ] \]