Transcript for ‘Initial Value Problem’


Problem 6-5: The standard Initial Value Problem for second order linear constant coefficient differential equations.

Typically, models the damped oscillator. This oscillator is driven. This problem comprises a significant portion of the ODE course. Let's solve it with the ODE Analyzer Assistant. That's a tool that can be accessed through the menu. [He goes to the ‘Tools’ menu and clicks the ‘ODE Analyzer’ option in the drop-down menu.] It's this tool here. We...launch it from here. We will have to get the differential equation into the assistant. It's easier to begin with the differential equation and use the context menu to access the assistant. [He highlights the differential equation with his mouse and drags it down to the second column of the ‘Solution by ODE Analyzer Assistant’ table]. So...I have dragged down a copy of the differential equation.

And let me pause here to indicate how the differential equation is written. 'y' and an apostrophe is understood by Maple to mean differentiate with respect to 'x', so if I hit the enter key you can see that Maple interprets the apostrophe as derivative with respect to 'x'. However, I prefer the look of these primes. They are in my favorites pallet. I got them from the punctuation pallet. You see there is a single, prime, double, triple and even a quadruple prime. I like how they look better than I like how the apostrophe looks. And especially if you have 'y' and two apostrophes, as opposed to 'y' and a double prime symbol. I just think the double prime symbol looks better.

So, the default on the prime is to differentiate with respect to 'x'. If you want to differentiate with respect to ’t’ as our problem requires, we could simply include 't' as the independent variable, and then differentiation is with respect to 't'. There is a way of setting that through...[He clicks on the ’View’ menu and chooses the ‘Typesetting Rules’ option from the drop-down menu.’] the Typesetting Rules Assistant to show you that the Typesetting Rules Assistant. You see down here it says 'Prime'. [Referring to the ‘Prime Derivatives’ check box in the ‘Differential Options’ section of the ‘Typesetting Rules Assistant’ window.] This is with respect to 'x'. Overdot is with respect to 't' [Referring to the ‘Dot Derivatives’ checkbox], and here is where you can actually modify that for a complete session[Referring to the ‘Differential Options’ section].

Here's our differential equation [in the second column of the ‘Solution by ODE Analyzer Assistant’ table]. The context menu provides two relevant choices [right clicks and chooses ‘Solve DE’ from the menu that appears]: Solve the DE. This would give the general solution with two arbitrary constants; and solve the DE Interactively [moves one down in the menu to the ‘Solve DE Interactively’ option], which will launch the assistant; the differential equation is already embedded. And we can interactively add either initial or boundary conditions. [ODE Analyzer Assistant appears in a new window. He clicks the ‘Edit’ button below the conditions column and a ‘Edit Conditions’ window opens.] This is an initial value problem, so it says 'y'… at 0. [He is in the ‘Add conditions’ section of the ‘Edit conditions’ window. He fills in each field and clicks the ‘Add’ button. The equation appears in one slot in the ‘Edit Conditions’ section. He then repeats the steps to add another condition]. We have the value 2. This is a drop-down. Y prime at 0 is -1. We have the two conditions. Click 'done'. Now, we can solve this problem either numerically or clearly symbolically. Let's try symbolically.

We have a solve button. Click the solve button. Maple gives us the analytics solution, which we can view in a larger window if we have to. And there is a plot button, which draws a graph of the solution. Now, we can modify this graph in several ways by clicking the 'Plot Options' button; for example, you click 'Copy'; you get a copy of 'y'. And let's change that to 'y' prime. [Edits the options in ‘Plot Options’ and then clicks the ‘Done’ button.] So, now when we click 'Plot' [in the ‘Solve symbolically’ window.] we get the solution in red and the derivative in green. We can even do things like this [clicks the ‘Plot Options’ button again. He goes to the ‘Axes and Independent variable range’ section of the ‘Plot Options’ window.]; the horizontal access...we will set to be the dependent variable 'y', but that is making 'y' the independent variable to the calculation. We will leave y prime as the dependent variable. And now when we click plot, we are seeing a trajectory in the phase plane. This check box [the ‘Show Maple Commands’ check box in the ‘Solve Symbolically’ window] will list the Maple commands in effect, so here you see the dsolve command which gave the solution. And down here you'll see the plot command, which draws the graph. When we click 'Quit' we can return the solution; we can return the plot; we can return the Maple commands; or we can return nothing at all.