Internal problem ID [14038]
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).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+3 y \left (t \right )+5 \operatorname {Heaviside}\left (-2+t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+6 y \left (t \right )+17 \operatorname {Heaviside}\left (-2+t \right ) \end {align*}
With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 67
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: 0.11 (sec). Leaf size: 60
DSolve[{x'[t]==4*x[t]+3*y[t]+5*UnitStep[t-2],y'[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\geq 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\geq 2 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \\ \end{align*}