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.)
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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.
I basically need help finding the rest of the solutions.
To find the general solution for the displacement of the heart tissue at time t > 0 in case of zero damping and assuming v = sqrt(4mk), we need to apply an alternative method since the quadratic equation yields repeated roots.
Here's how you can proceed:
1. Let's consider the case where the roots of the characteristic equation are r1 = r2 = (sqrt(mk))/(m).
2. In this case, we need to modify our general solution form to include an extra term. The general solution will be of the form: y(t) = (C1 + C2 * t) * e^((sqrt(mk))/(m) * t), where C1 and C2 are arbitrary constants.
3. To find the values of C1 and C2, we need additional conditions or information. These can be initial conditions or specific values of displacement and velocity at a given time. Without those conditions, we cannot determine the exact values of C1 and C2, so we leave them as arbitrary constants.
Therefore, the general solution for the displacement of the heart tissue at time t > 0, under the given conditions, is:
y(t) = (C1 + C2 * t) * e^((sqrt(mk))/(m) * t)
Note that the arbitrary constants C1 and C2 will vary based on specific conditions or constraints given in the problem or experiment setup.