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Sturm-Liouville - Problem

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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

Exercise 3:

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.

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Solution Summary

This shows how to recast a given equation in Sturm-Liouville form. It is answered in step by step break downs and equations.

Solution Preview

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 ...

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