Question

Diff EQ: Show that y_1(t) = t^(-1/2)sin(t) and y_2(t) = t^(-1/2)cos(t) are solutions to the homogeneous problem x^2y'' + xy' + (x^2 - .25)y = g(x). Also find a particular solution to the non-homogeneous problem where g(x) is an arbitrary cont. function.

Answer 1

replace t with x in the first sentence

Answer 2

Well for the first part, just plug them into the equation, replacing g(x) with 0, and demonstrate that it works.

Answer 3

I thought thats what was going to happen for the first part, what about the second part?. Not sure about the non-constant coefficients.

Answer 4

@dan815 @jim_thompson5910 @whpalmer4

Answer 5

Just to clarify: Do you mean \(g(x)\) is a continuous or constant function?

Answer 6

@satellite73 ??

Answer 7

g(x) is continuous

Answer 8

@Directrix

Answer 9

Maybe this will help? I am really tired right now and cannot think straight.
http://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx