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the Differential Equation u"(x) − u(x) = f(x) |
The differential equation
is a linear inhomogeneous ordinary differential equation. Its solution can be obtained most easily using Fourier or Laplace Transforms.
Taking the Laplace Transform of the above equation gives:
where U is the Laplace transform of u(x) and F is the Laplace transform of f(x) and where it is assumed that u(0)=0 and u'(0)=0.
This means that
The Laplace transform of sinh(x) is equal to −½[1/(S+1) - 1/(S-1)] so
By the convolution theorem then
To verify that this a solution, first
The second derivative is given by
The lower limit of 0 for the integral is arbitrary, stemming from the way the Laplace transform has to be defined. It is appropriate to take the lower limit as −∞ and to take u(−∞)=0 and u'(−∞)=0. Thus the solution is
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