Construct Green's function for the differential equation
under the conditions that y(0) is bounded and .
Based on the image provided, here is the transcription of the mathematical problems:
The given differential equation is: [ x y'' + y' = 0, \quad 0 < x < \ell ] with boundary conditions: (y(0)) bounded and (y(\ell) = 0). --- Step 1: Solve the homogeneous equation The equation can be rewritten as: [ y'' + \frac{1}{x} y' = 0. ] Let (y' = u). Then (y'' = u'), and the equation becomes: [ u' + \frac{1}{x} u = 0. ] _________ _______ _____ ______ ____ ________ _________ __________.
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(i) , (ii)
, solve the initial value problem
Upto . If the exact solution is
, obtain the error.
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y = x - 2y, y(0) = 1
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and
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Based on the third image provided, here is the transcription of the remaining mathematical problems:
Find the Laplace transform of
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.
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