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Applied Math (Biological application)

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The deformation y(t) of heart muscle from its rest position after t seconds can be modelled as a spring-mass system, described by the differential equation:

my" + vy' + ky = F(t)

where m, v, and k are positive constants denoting the mass of the myocardial tissue, and the stiffness and damping coefficients, respectively; F(t) is called a forcing function.

1. In case of zero and assuming v = sqrt(4mk) ; find the general solution for the displacement of the heart tissue at time t > 0 (your answer will contain m, k, t, and arbritrary constants.)


Alrighty so this is how I chose to attempt the question:

1. First I converted the equation to auxillary form, meaning the y" became r^2, y' became r, and y became 1. This gives me mr^2 + vr + k. I make the right-hand side = to 0 and solve using the quadratic formula. What I get is r = (sqrt(mk))/(m). This yields the solution y=e^((sqrt(mk))/(m)x).

I am having to do an alt. method for this question because it is considered an example of "repeated roots" (look at outcome of quadratic equation) so I am now having to do a full out different process from usual. Can anybody help out here? Thanks.

  • Applied Math (Biological application) -

    I basically need help finding the rest of the solutions.

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