73.27.18 problem 38.10 (L)

Internal problem ID [15774]
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 : Tuesday, January 28, 2025 at 08:06:58 AM
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*}

✓ Solution by Maple

Time used: 0.066 (sec). Leaf size: 66

dsolve([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), x(0) = 0, y(0) = 0], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.112 (sec). Leaf size: 60

DSolve[{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]},{x[0]==0,y[0]==0},{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*}