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.

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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.

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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replace t with x in the first sentence
Well for the first part, just plug them into the equation, replacing g(x) with 0, and demonstrate that it works.
I thought thats what was going to happen for the first part, what about the second part?. Not sure about the non-constant coefficients.

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Just to clarify: Do you mean \(g(x)\) is a continuous or constant function?
g(x) is continuous
Maybe this will help? I am really tired right now and cannot think straight. http://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx

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