67.27.14 problem 38.10 (h)

Internal problem ID [17057]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.10 (h)
Date solved : Thursday, October 02, 2025 at 01:42:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=8 x \left (t \right )+2 y \left (t \right )+7 \,{\mathrm e}^{2 t}\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )+y \left (t \right )-7 \,{\mathrm e}^{2 t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.170 (sec). Leaf size: 23

ode:=[diff(x(t),t) = 8*x(t)+2*y(t)+7*exp(2*t), diff(y(t),t) = 4*x(t)+y(t)-7*exp(2*t)]; 
ic:=[x(0) = -1, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -\frac {3}{2}+\frac {{\mathrm e}^{2 t}}{2} \\ y \left (t \right ) &= 6-5 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 28

ode={D[x[t],t]==8*x[t]+2*y[t]+7*Exp[2*t],D[y[t],t]==4*x[t]+y[t]-7*Exp[2*t]}; 
ic={x[0]==-1,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} \left (e^{2 t}-3\right )\\ y(t)&\to 6-5 e^{2 t} \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-8*x(t) - 2*y(t) - 7*exp(2*t) + Derivative(x(t), t),0),Eq(-4*x(t) - y(t) + 7*exp(2*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1}}{4} + 2 C_{2} e^{9 t} + \frac {e^{2 t}}{2}, \ y{\left (t \right )} = C_{1} + C_{2} e^{9 t} - 5 e^{2 t}\right ] \]