How to Solve an IVP Involving Dirac Delta Function?

In summary: In particular \int_{t= a}^{t= b} k\delta(t- a)dt= k.In summary, the Dirac Delta function is a mathematical tool used to represent an instantaneous impulse or "jerk" in a system. It is commonly used in differential equations to model situations where a sudden change occurs. By solving the initial value problem and using the solution, the constant H can be explained as the magnitude of the displacement caused by the impulse. Finally, by choosing a specific value for H, the prescribed displacement from equilibrium can be achieved at a given time t greater than or equal to a.
  • #1
mango84
4
0
Dirac Delta Function:

If, at time t =a, the upper end of an undamped spring-mass system is jerked upward suddenly and returned to its original position, the equation that models the situation is mx'' + kx = kH delta(t-a); x(0) = x(sub zero), x'(0) = x(sub 1), where m is the mass, k is the spring constant, and H is a constant.

(a) Solve the IVP manually, with x(0)=0=x'(0)

(b) Use the solution found in part (a) to explain the significance of the constant H.

(c) Choose a value for H such that the mass achieves a prescribed displacement from equilibrium A for t (greater than or equal to) a.

Does anybody know how to do this? I'm lost!
 
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  • #2
Pretty much just "go ahead and do it"! You certainly should be able to find the general solution to the corresponding "homogenous equation", mx'' + kx = 0. Now use "variation of parameters" to find a solution to the entire equation. You may remember that that involves looking for a solution of the form x(t)= u(x)y1(x)+ v(x)y2(x) where y1 and y2 are two independent solutions to the homogeneous equation and u and v are functions you need to find. Differentiating x'= u'y1+ u y1'+ v' y2+ v y2'. "Narrow the search" by requiring that the terms involving u' and v' to be 0: u'y1+ v'y2= 0. Now x'= uy1'+ vy2' so differetiating again, x"= u'y1+ uy1''+ v'y2+ vy2''. Putting those x'', x', and x'' you see that, because y1 and y2 are solutions to the homogenous equation, anything not involving u' and v' cancel. You get an equation involving only u', v', and, of course the right hand side of the differential equation. That together with u'y1+ v'y2= 0 gives you two equations you can solve algebrically for u' and v'. Finding u and v then involves integrating those. The whole point of the delta function is that [itex]\int f(x)\delta(x)dx= f(0)[/itex] as long as the interval of integration includes 0.
 

What is the Dirac Delta function?

The Dirac Delta function, commonly denoted as δ(x), is a mathematical construct used in many areas of physics and engineering. It is defined as an infinitely tall and infinitely narrow spike at x = 0, with an area of 1 under the curve. In other words, it is a function that is 0 everywhere except at x = 0, where it is infinite.

What are some common applications of the Dirac Delta function?

The Dirac Delta function has a wide range of applications, including in signal processing, control systems, quantum mechanics, and electromagnetism. It is often used to model impulsive forces, point charges, and point masses.

How is the Dirac Delta function different from a regular function?

The Dirac Delta function is not a regular function in the traditional sense. It cannot be graphed or evaluated at specific points, and its integral over any interval is always equal to 1. Additionally, it is a distribution rather than a function, meaning it is defined in terms of its behavior when integrated with another function.

What is the relationship between the Dirac Delta function and the Kronecker Delta function?

The Dirac Delta function and the Kronecker Delta function are closely related, but they serve different purposes. The Dirac Delta function is defined for continuous variables, while the Kronecker Delta function is defined for discrete variables. In essence, the Kronecker Delta function is the discrete version of the Dirac Delta function.

How is the Dirac Delta function used in solving differential equations?

The Dirac Delta function is often used in solving differential equations because it can be used to model impulsive forces or point masses. This allows for the solution of certain differential equations that would otherwise be difficult to solve. Additionally, the Dirac Delta function can be used to represent initial conditions or boundary conditions in some cases.

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