How to Solve an IVP Involving Dirac Delta Function?

[mango84]

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!

 

[HallsofIvy]

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 \int f(x)\delta(x)dx= f(0) as long as the interval of integration includes 0.