67.27.18 problem 38.10 (L)

Internal problem ID [17061]
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 (L)
Date solved : Thursday, October 02, 2025 at 01:42:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+3 y \left (t \right )+5 \operatorname {Heaviside}\left (t -2\right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+6 y \left (t \right )+17 \operatorname {Heaviside}\left (t -2\right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.114 (sec). Leaf size: 66

ode:=[diff(x(t),t) = 4*x(t)+3*y(t)+5*Heaviside(t-2), diff(y(t),t) = x(t)+6*y(t)+17*Heaviside(t-2)]; 
ic:=[x(0) = 0, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 2 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-14+7 t}-3 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-6+3 t}+\operatorname {Heaviside}\left (t -2\right ) \\ y \left (t \right ) &= 2 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-14+7 t}+\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-6+3 t}-3 \operatorname {Heaviside}\left (t -2\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 60

ode={D[x[t],t]==4*x[t]+3*y[t]+5*UnitStep[t-2],D[y[t],t]==x[t]+6*y[t]+17*UnitStep[t-2]}; 
ic={x[0]==0,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} 1+2 e^{7 (t-2)}-3 e^{3 t-6} & t>2 \\ 0 & \text {True} \\ \end {array} \\ \end {array}\\ y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} -3+2 e^{7 (t-2)}+e^{3 t-6} & t>2 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.362 (sec). Leaf size: 114

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) - 3*y(t) - 5*Heaviside(t - 2) + Derivative(x(t), t),0),Eq(-x(t) - 6*y(t) - 17*Heaviside(t - 2) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 C_{1} e^{3 t} + C_{2} e^{7 t} + \frac {2 e^{7 t} \theta \left (t - 2\right )}{e^{14}} - \frac {3 e^{3 t} \theta \left (t - 2\right )}{e^{6}} + \theta \left (t - 2\right ), \ y{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{7 t} + \frac {2 e^{7 t} \theta \left (t - 2\right )}{e^{14}} + \frac {e^{3 t} \theta \left (t - 2\right )}{e^{6}} - 3 \theta \left (t - 2\right )\right ] \]