14.20.10 problem 10

Internal problem ID [2707]
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 : 10
Date solved : Tuesday, March 04, 2025 at 02:40:05 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-4 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*}

With initial conditions

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 31

ode:=[diff(x(t),t) = 3*x(t)-4*y(t)+exp(t), diff(y(t),t) = x(t)-y(t)+exp(t)]; 
ic:=x(0) = 1y(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (-t^{2}-t +1\right ) \\ y &= \frac {{\mathrm e}^{t} \left (-2 t^{2}+4\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 31

ode={D[x[t],t]==3*x[t]-4*y[t]+Exp[t],D[y[t],t]==x[t]-y[t]+Exp[t]}; 
ic={x[0]==1,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.141 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 4*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)
 

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