76.6.1 problem 1

Internal problem ID [17385]
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 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 1
Date solved : Monday, March 31, 2025 at 04:12:32 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+4 \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 31

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = x(t)+4]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_2 +{\mathrm e}^{t} c_1 -4 \\ y \left (t \right ) &= -{\mathrm e}^{-t} c_2 +{\mathrm e}^{t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 70

ode={D[x[t],t]==y[t],D[y[t],t]==x[t]+4}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (-8 e^t+(c_1+c_2) e^{2 t}+c_1-c_2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(-x(t) + Derivative(y(t), t) - 4,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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