Reduction of order certainly should work. Exactly what have you tried?

If then and .

Putting that into the equation, [tex]= u"t^{1/2}+ u't^{-1/2}= f cos(t)[tex]. Let v= u' and that is ][tex]v't^{1/2}+ vt^{-1/2}= f cos(t)[tex], a linear first order equation. You might have to leave part of the solution in terms of an integral.