65.14.4 problem 26.1 (iv)

Internal problem ID [13741]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 26, Explicit solutions of coupled linear systems. Exercises page 257
Problem number : 26.1 (iv)
Date solved : Wednesday, March 05, 2025 at 10:14:45 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )-4 y \left (t \right )+{\mathrm e}^{3 t}\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 34

ode:=[diff(x(t),t) = 5*x(t)-4*y(t)+exp(3*t), diff(y(t),t) = x(t)+y(t)]; 
ic:=x(0) = 1y(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (t^{2}+7 t +1\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (t^{2}+6 t -2\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 39

ode={D[x[t],t]==5*x[t]-4*y[t]+Exp[3*t],D[y[t],t]==x[t]+y[t]}; 
ic={x[0]==1,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{3 t} \left (t^2+7 t+1\right ) \\ y(t)\to \frac {1}{2} e^{3 t} \left (t^2+6 t-2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) + 4*y(t) - exp(3*t) + Derivative(x(t), t),0),Eq(-x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = t^{2} e^{3 t} + t \left (2 C_{1} + 1\right ) e^{3 t} + \left (C_{1} + 2 C_{2}\right ) e^{3 t}, \ y{\left (t \right )} = C_{1} t e^{3 t} + C_{2} e^{3 t} + \frac {t^{2} e^{3 t}}{2}\right ] \]