76.6.13 problem 15

Internal problem ID [17397]
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 : 15
Date solved : Monday, March 31, 2025 at 04:12:48 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+y \left (t \right )+1\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right )-3 \end{align*}

✓ Maple. Time used: 0.118 (sec). Leaf size: 69

ode:=[diff(x(t),t) = -x(t)+y(t)+1, diff(y(t),t) = x(t)+y(t)-3]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\sqrt {2}\, t} c_2 +{\mathrm e}^{-\sqrt {2}\, t} c_1 +2 \\ y \left (t \right ) &= \sqrt {2}\, {\mathrm e}^{\sqrt {2}\, t} c_2 -\sqrt {2}\, {\mathrm e}^{-\sqrt {2}\, t} c_1 +{\mathrm e}^{\sqrt {2}\, t} c_2 +{\mathrm e}^{-\sqrt {2}\, t} c_1 +1 \\ \end{align*}

✓ Mathematica. Time used: 0.215 (sec). Leaf size: 160

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

\begin{align*} x(t)\to \frac {1}{4} e^{-\sqrt {2} t} \left (8 e^{\sqrt {2} t}+\left (\sqrt {2} c_2-\left (\sqrt {2}-2\right ) c_1\right ) e^{2 \sqrt {2} t}+\left (2+\sqrt {2}\right ) c_1-\sqrt {2} c_2\right ) \\ y(t)\to \frac {1}{4} e^{-\sqrt {2} t} \left (4 e^{\sqrt {2} t}+\left (\sqrt {2} c_1+\left (2+\sqrt {2}\right ) c_2\right ) e^{2 \sqrt {2} t}-\sqrt {2} c_1-\left (\sqrt {2}-2\right ) c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.318 (sec). Leaf size: 61

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

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