82.4.21 problem 28-21

Internal problem ID [21839]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 28. Laplace transforms. Page 850
Problem number : 28-21
Date solved : Thursday, October 02, 2025 at 08:02:44 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-6 x \left (t \right )+3 y&=8 \,{\mathrm e}^{t}\\ y^{\prime }-2 x \left (t \right )-y&=4 \,{\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: 0.074 (sec). Leaf size: 41

ode:=[diff(x(t),t)-6*x(t)+3*y(t) = 8*exp(t), diff(y(t),t)-2*x(t)-y(t) = 4*exp(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -4 \,{\mathrm e}^{3 t}+7 \,{\mathrm e}^{4 t}-2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -4 \,{\mathrm e}^{3 t}+\frac {14 \,{\mathrm e}^{4 t}}{3}-\frac {2 \,{\mathrm e}^{t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 51

ode={D[x[t],t]-6*x[t]+3*y[t]==8*Exp[t],D[y[t],t]-2*x[t]-y[t]==4*Exp[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^t \left (-4 e^{2 t}+7 e^{3 t}-2\right )\\ y(t)&\to \frac {2}{3} e^t \left (-6 e^{2 t}+7 e^{3 t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-6*x(t) + 3*y(t) - 8*exp(t) + Derivative(x(t), t),0),Eq(-2*x(t) - y(t) - 4*exp(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 7 e^{4 t} - 4 e^{3 t} - 2 e^{t}, \ y{\left (t \right )} = \frac {14 e^{4 t}}{3} - 4 e^{3 t} - \frac {2 e^{t}}{3}\right ] \]