How to Solve y''+4y = x*sin2x Using Undetermined Coefficients?

[kasse]

y''+4y = x*sin2x

Find a particular solution using the method of undetermined coefficients.
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No idea what to do here. My table has no suggestion for a particular solution when r(x) = x*sin2x.

 

[rock.freak667]

The RHS consists of the product of a Polynomial of degree one and a trig function

so the P.I. would be (P.I.of the polynomial)*(P.I. for the trig function)

 

[HallsofIvy]

y''+4y = x*sin2x

Find a particular solution using the method of undetermined coefficients.
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No idea what to do here. My table has no suggestion for a particular solution when r(x) = x*sin2x.

You don't learn mathematics by using tables, you learn mathematics by using concepts. For sin(2x) on the right hand side your first thought should be Acos(2x)+ Bsin(2x). With "x" you would try Ax+ B. As rockfreak667 said, combine those ideas:
try y= (Ax+B)sin(2x)+ (Cx+ D)cos(2x)

HOWEVER, sin(2x) and cos(2x) already satisfy the differential equation! I'm sure you already know what to do in that situation- multiply by x. Try (Ax²+ Bx)sin(2x)+ (Cx²+ Dx)cos(2x).

I suspect that, since the equation only involves "even" derivatives, you don't need all of those coefficients- some will be 0- but I see no good way to figure out which in advance.