14.26.9 problem 9

Internal problem ID [2782]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 3. Systems of differential equations. Section 3.13 (Solving systems by Laplace transform). Page 370
Problem number : 9
Date solved : Monday, January 27, 2025 at 06:13:47 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )+\delta \left (t -\pi \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.097 (sec). Leaf size: 58

dsolve([diff(x__1(t),t) = 2*x__1(t)-2*x__2(t), diff(x__2(t),t) = 4*x__1(t)-2*x__2(t)+Dirac(t-Pi), x__1(0) = 1, x__2(0) = 0], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \sin \left (2 t \right )+\cos \left (2 t \right )-\sin \left (2 t \right ) \operatorname {Heaviside}\left (t -\pi \right ) \\ x_{2} \left (t \right ) &= -\sin \left (2 t \right ) \operatorname {Heaviside}\left (t -\pi \right )+\cos \left (2 t \right ) \operatorname {Heaviside}\left (t -\pi \right )+2 \sin \left (2 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 54

DSolve[{D[x1[t],t]==2*x1[t]-2*x2[t]+0,D[x2[t],t]==4*x1[t]-2*x2[t]+DiracDelta[t-Pi]},{x1[0]==1,x2[0]==0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -\theta (t-\pi ) \sin (2 t)+\sin (2 t)+\cos (2 t) \\ \text {x2}(t)\to \theta (t-\pi ) (\cos (2 t)-\sin (2 t))+2 \sin (2 t) \\ \end{align*}