Diff Eq Question - Impulse Functions

In summary, the conversation discusses a system in Example 1 from Boyce and DiPrima's Elementary Differential Equations (9th Ed.), where the goal is to bring the system to rest after exactly one cycle. Part (a) asks for the impulse needed to accomplish this objective, and the solution is given as -e^{\frac{-T}{4}}\delta(t-5-T), where T=\frac{8\pi}{\sqrt{15}}. Part (b) requires solving the resulting initial value problem and confirming that it behaves as specified. The total response of the system is the sum of individual responses, using the principle of superposition.
  • #1

Homework Statement



From Boyce and DiPrima's Elementary Differential Equations (9th Ed.), Section 6.5, Problem 13:

Consider again the system in Example 1:

[itex]2y'' + y' + 2y = \delta(t-5); y(0)=y'(0)=0[/itex]

(Solution in text: [itex]y=\frac{2}{\sqrt{15}}u_{5}(t)e^{\frac{-(t-5)}{4}}sin\frac{\sqrt{15}}{4}(t-5)[/itex])

Suppose that it is desired to bring the system to rest again after exactly one cycle - that is, when the response first returns to equilibrium moving in the positive direction.

(a) Determine the impulse [itex]k \delta(t-t_{0})[/itex] that should be applied to the system in order to accomplish this objective. Note that k is the magnitude of the impulse and [itex]t_{0}[/itex] is the time of its application.

(b) Solve the resulting initial value problem and plot its solution to confirm that it behaves in the specified manner.

Homework Equations



The solution to part (a) given at the back of the text is

[itex]-e^{\frac{-T}{4}}\delta(t-5-T), T=\frac{8\pi}{\sqrt{15}}[/itex]

The Attempt at a Solution



Honestly, I'm completely lost here.

I get why T must equal 8pi/sqrt(15) - we need the stuff inside sin(blah) to equal 2pi since that will tell us when one cycle is completed; this happens when t-5=8pi/sqrt(15). But I'm confused as to how that gets us to the answer provided above and, moreover, how to proceed with part (b).

Any help would be appreciated.
 
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  • #2
The total response of the system is the sum of the individual responses to each impulse. This is just the principle of superposition. Write down the response to ##k\delta(t-t_0)## and add it to the response from the first impulse and set the sum equal to 0.
 

1. What is an impulse function in differential equations?

An impulse function, also known as a Dirac delta function, is a mathematical concept used to model a sudden, instantaneous change in a system. It is represented by the symbol δ(t) and has a value of 0 for all values of t except at t = 0, where it has a value of infinity. In differential equations, it is used to represent an instantaneous change in a variable.

2. How do you solve differential equations with impulse functions?

When solving differential equations with impulse functions, you can treat the impulse function as a regular function and solve the equation as you normally would. However, when you reach the time point at which the impulse function occurs, you must apply the definition of the impulse function, which states that the integral of the impulse function over an interval containing t = 0 is equal to 1. This will help you determine the value of the variable at t = 0.

3. Can impulse functions be used to model real-world phenomena?

Yes, impulse functions can be used to model real-world phenomena, such as a sudden change in temperature or pressure, an impact or collision, or the firing of a neuron in the brain. They are also commonly used in control systems and signal processing to represent sudden changes in input or output signals.

4. What is the Laplace transform of an impulse function?

The Laplace transform of an impulse function is 1. This can be easily derived from the definition of the Laplace transform, which involves taking the integral of a function multiplied by e^(-st) over the interval from 0 to infinity. When this is applied to the impulse function, it simplifies to 1.

5. Are impulse functions related to delta functions in other areas of mathematics?

Yes, impulse functions are closely related to delta functions in other areas of mathematics, such as Fourier analysis and distribution theory. In these areas, the Dirac delta function is defined in a similar way as in differential equations, but it is used to represent a distribution or generalized function rather than a point of discontinuity in a function.

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