14.20.4 problem 4

Internal problem ID [2701]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 4
Date solved : Tuesday, March 04, 2025 at 02:39:58 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 77

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

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

✓ Mathematica. Time used: 0.383 (sec). Leaf size: 164

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

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

✓ Sympy. Time used: 0.414 (sec). Leaf size: 65

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

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