How to Solve the Differential Equation x'' - x = t sin(t)?

rbailey5
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Homework Statement


x(t)^double prime-x(t)=tsint


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The Attempt at a Solution



in this case would I start this as x=t(A(2)+A(1))sint or do i also have to use the cos which makes it
x=t(A(2)+A(1))sint+t(B(2)+B(1))cost?
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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