Use Heaviside's expansion formula derived in Problem 40 to determine the inverse Laplace transform of
Given a function , if there is a function that is continuous on
and satisfies,then we say that is the inverse Laplace transform of and employ the notation
Non-repeated Linear Factors
If can be factored into a product of distinct linear factors,
where the 's are all distinct real numbers, then the partial fraction expansion has the form
where the 's are real numbers. There are various ways of determining the constants . In the next example, we demonstrate two such methods.2.
Repeated Linear Factors
If is a factor of and is the highest power of that divides , then the portion of the partial fraction expansion of that corresponds to the term is
where the 's are real numbers.
If is a quadratic factor of that cannot be reduced to linear factors with real coefficients and is the highest power of that divides , then the portion of the partial fraction expansion that corresponds to is
it is more convenient to express in the form when we look up the Laplace transforms. So let's agree to write this portion of the partial fraction expansion in the equivalent form
There we have
Now we have
Using the equation from Problem 40 we get
Nonlinear Spring. The Duffing equation where r is a constant is a model for the vibrations of amass attached to a nonlinear spring. For this model, does the period of vibration vary as the parameter r is varied?
Does the period vary as the initial conditions are varied? [Hint: Use the vectorized Runge–Kutta algorithm with h = 0.1 to approximate the solutions for r = 1 and 2,
with initial conditions for a = 1, 2, and 3.]
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