"If the fundamental set for a given differential equation is {1, x, cos(2x), sin(2x)} and the right hand side function is g(x) = 1 + 2x + x sin(2x), state the trial function you would maketo find yp using the method of undetermined coefficients." ------------------------------------------------- Okay so based on this question, yc=c1+c2ex+c3cos(2x)+c4sin(2x) right? But I'm confused about the trial function for yp. For 1+2x, we would use Ax+B, but would the B component be considered a duplicate in this case (duplicate of c1 in yc)? If it is infact a duplicate term, would we multiply the entirety of 1+2x by an additional x? Or just the B? Now for xsin(2x), the trial function should be (Cx+D)cos(2x)+(Ex+F)sin(2x) right? Are these terms considered duplicates of c3cos(2x)+c4sin(2x)?? Why or why not, please help me as I'm very confused, and state the rule for considering a term to be a duplicate.  What should my final answer be? Thank you so much for all your help :)

"If the fundamental set for a given differential equation is {1, x, cos(2x), sin(2x)} and the right hand side function is g(x) = 1 + 2x + x sin(2x), state the trial function you would make
to find y_p using the method of undetermined coefficients."

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Okay so based on this question, y_c=c₁+c₂e^x+c₃cos(2x)+c₄sin(2x) right?

But I'm confused about the trial function for y_p. For 1+2x, we would use Ax+B, but would the B component be considered a duplicate in this case (duplicate of c₁ in y_c)? If it is infact a duplicate term, would we multiply the entirety of 1+2x by an additional x? Or just the B?

Now for xsin(2x), the trial function should be (Cx+D)cos(2x)+(Ex+F)sin(2x) right? Are these terms considered duplicates of c₃cos(2x)+c₄sin(2x)?? Why or why not, please help me as I'm very confused, and state the rule for considering a term to be a duplicate. 

What should my final answer be? Thank you so much for all your help :)