Laplace Transform Videos

Question 1

\displaystyle \ddot{x} + 9x = \delta(t-1) , \ \ x(0) = 0, \ \dot{x}(0) = 0

Solve the above IVP using Laplace Transforms and then find

x(2.0)

Answer to Question 1:   

x(2.0)   = \underline{\ \hspace{.75in}} .

See graph and video below  (in 2 parts) for solution.  Total run time about 10 minutes.
Part 2 has the final answer.

 

Part 1.

Video Player

 

 

Part 2.

Video Player

 

 

Notes.
The Heaviside function is the unit step function u(t).
The Dirac function is \delta(t).

Below are the Maple commands to solve the IVP in Question 1 and create the above Figure.  You can copy and paste these commands into Maple.

1
2
3
4
5
6
7
8
9
10
de := diff(x(t), t $ 2) + 4*x(t) = Dirac(t - 2);
tq := 2.5;
dsolve(de, x(t));
ics := x(0) = 0, D(x)(0) = 0;
SolutionIVP := dsolve({de, ics});
x(tq) = evalf(subs(t = tq, rhs(SolutionIVP)));
plot(rhs(SolutionIVP), t = 0 .. 5, x = -0.5 .. 0.5, thickness = 4,
  size = [0.5, 0.6], gridlines,
  title = typeset("Graph of the solution x(t) of \n", de, " with IC ", ics),
  caption = typeset(SolutionIVP));

Below is a screen capture of above commands in Maple.

 

Question 2

\displaystyle \ddot{x} - 5\dot{x} + 6x = \delta(t-4), \ \ x(0) = 5, \ \dot{x}(0) = 7

Solve the above IVP using Laplace Transforms and then find

x(4.5)

Answer to Question 2:   

x(4.5)   = \underline{\ \hspace{.75in}} .

See video below  (in 5 parts) for solution.  Total run time about 30 minutes.  Part 5 has the final answer and shows how to solve this question using Maple.

 

Part 1.

Video Player

 

 

Part 2.

Video Player

 

 

Part 3.

Video Player

 

 

Part 4.

Video Player

 

 

Part 5.

Video Player

 

 

 

 

 

Attribution-NonCommercial-ShareAlike 4.0 International

This entry is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International license.