4. Use the results of Exercise 3 to recast each of the following differential equations in the Sturm-Liouville form (1a). Identify p(x), q(x), and w(x).
(a) xy" + 5y' + lambda xy = 0
(b) y" + 2y' + xy + lambda x^2y = 0
(c) y" + y' + lambda y = 0
(d) y" - y' + lambda xy = 0
(e) x^2y" + xy' + lambda x^2y = 0
(f) y" + (cot x)y' + lambda y = 0
3. (Obtaining Sturm-Liouville form) We observed, in Example 4, that the equation
A(x)y" + B(x)y' + C(x)y + lambda D(x)y = 0 (3.1)
is in the standard Sturm-Liouville form (1a) only if B(x) = A'(x). Show that if A(x) =/= 0 on [a, b] and (B - A')/A is continuous on [a, b], then we can recast (3.1) in the form (1a) by multiplying (3.1) by
sigma(x) = e^(integral[(B-A')/A]dx
Please solve for only part b.© BrainMass Inc. brainmass.com October 16, 2018, 5:05 pm ad1c9bdddf
Ok, in part (b), comparing to what we have in the general case:
A(x)=1, B(x)=2, C(x)=x and D(x)=x^2.
Now we form (B-A')/A= (2-0)/1=2
Therefore if we ...
This shows how to recast a given equation in Sturm-Liouville form. It is answered in step by step break downs and equations.
Eigenvalues, Eigenfunctions and Sturm-Liouville Problems
1. y'' + k*y = 0 BC: y'(0) = 0 y'(L) = 0
2. y'' + k*y = 0 BC: y(0) = y() y'(0) = y'()
3. y'' + k*y = 0 BC: y(0) = 0 y() +2*y'() = 0
4. y'' + 2*y' + (1+k)*y = 0 BC: y(0) = y(1) =0
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