Graphing a d.e. w/ a discountinous forcing function

Dec 1, 2005

georgeh

hey all
i have the following d.e.
y''+y=f(t); y(0)=0, y'(0)=1;
f(t)={ 1 , 0<=t < pi/2
{ 0, pi/2 <=t < infinity
so i solved the d.e.
and i got
Y(t) = 1+sin-cost-U_pi/2[1-sint]
i did all the work, i just don't know how to go about graphing the funciton..
any help would be appreciated.

Dec 1, 2005

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

I'm having great difficulty trying to figure out what you mean in your expression for Y(t).

1. How do you graph a differential equation with a discontinuous forcing function?

To graph a differential equation with a discontinuous forcing function, you first need to identify the points at which the function is discontinuous. Then, you need to break the graph into separate pieces at those points. For each piece, you can solve the differential equation just as you would for a continuous function and plot the resulting curve. However, at the points of discontinuity, you will need to use the limit of the function to determine the value of the derivative.

2. What is a discontinuous forcing function?

A discontinuous forcing function is a function in a differential equation that has at least one point of discontinuity. This means that the function is not continuous at that point and may have a jump or break in its graph. This can make solving and graphing the differential equation more complicated as it requires special considerations at the points of discontinuity.

3. Can a differential equation with a discontinuous forcing function have a solution?

Yes, a differential equation with a discontinuous forcing function can have a solution. However, it may require special techniques to solve and graph the equation accurately, as the points of discontinuity can affect the behavior of the solution. It is important to carefully identify and analyze the points of discontinuity in order to accurately solve and graph the equation.

4. How do you account for the points of discontinuity when graphing a differential equation?

To account for the points of discontinuity when graphing a differential equation, you can use the limit of the function at those points to determine the value of the derivative. This will allow you to plot the appropriate points on the graph and accurately represent the behavior of the function. It is important to carefully analyze the points of discontinuity and use the appropriate limits in order to accurately graph the differential equation.

5. Are there any special considerations when solving a differential equation with a discontinuous forcing function?

Yes, there are special considerations when solving a differential equation with a discontinuous forcing function. As mentioned earlier, the points of discontinuity require the use of limits to determine the value of the derivative. Additionally, the behavior of the solution may be affected by the points of discontinuity, so it is important to carefully analyze the function and its behavior at these points. It may also be necessary to break the graph into separate pieces and solve each piece separately in order to accurately represent the solution to the differential equation.