78.5.23 problem 8.b

Internal problem ID [21062]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 8.b
Date solved : Thursday, October 02, 2025 at 07:01:53 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+3 y+1\\ y^{\prime }&=-6 x \left (t \right )-4 y+{\mathrm e}^{t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.186 (sec). Leaf size: 43

ode:=[diff(x(t),t) = 5*x(t)+3*y(t)+1, diff(y(t),t) = -6*x(t)-4*y(t)+exp(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{-t}}{2}+4 \,{\mathrm e}^{2 t}-2-\frac {3 \,{\mathrm e}^{t}}{2} \\ y \left (t \right ) &= -{\mathrm e}^{-t}-4 \,{\mathrm e}^{2 t}+2 \,{\mathrm e}^{t}+3 \\ \end{align*}

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 52

ode={D[x[t],t]==5*x[t]+3*y[t]+1,D[y[t],t]==-6*x[t]-4*y[t]+Exp[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} \left (e^{-t}-3 e^t+8 e^{2 t}-4\right )\\ y(t)&\to -e^{-t}+2 e^t-4 e^{2 t}+3 \end{align*}

✓ Sympy. Time used: 0.192 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) - 3*y(t) + Derivative(x(t), t) - 1,0),Eq(6*x(t) + 4*y(t) - exp(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 4 e^{2 t} - \frac {3 e^{t}}{2} - 2 + \frac {e^{- t}}{2}, \ y{\left (t \right )} = - 4 e^{2 t} + 2 e^{t} + 3 - e^{- t}\right ] \]