76.10.5 problem 5

Internal problem ID [17456]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 5
Date solved : Monday, March 31, 2025 at 04:14:12 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -3 \end{align*}

✓ Maple. Time used: 0.116 (sec). Leaf size: 33

ode:=[diff(x(t),t) = 2*y(t), diff(y(t),t) = 8*x(t)]; 
ic:=x(0) = 1y(0) = -3; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= \frac {5 \,{\mathrm e}^{-4 t}}{4}-\frac {{\mathrm e}^{4 t}}{4} \\ y \left (t \right ) &= -\frac {5 \,{\mathrm e}^{-4 t}}{2}-\frac {{\mathrm e}^{4 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 40

ode={D[x[t],t]==2*y[t],D[y[t],t]==8*x[t]}; 
ic={x[0]==1,y[0]==-3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -\frac {1}{4} e^{-4 t} \left (e^{8 t}-5\right ) \\ y(t)\to -\frac {1}{2} e^{-4 t} \left (e^{8 t}+5\right ) \\ \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 34

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

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