ODE ( 2nd order nonhomogeneous equation)

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1. The problem statement, all variables and given/known data
By using the method of undetermined coefficients,find the particular solution of
y''+y'+y=(sin x)^2


2. Relevant equations
i know how to determine the particular solution IF it is sin x.
Ex: sin x ====> Asin x + B cos x (particular)

but i wonder how to determine the (sin x)^2


3. The attempt at a solution
 
i haven't actually seen a problem like this come up, but i think similarly to finding the particular solution to something like lhs=t^2 is A+Bt+Ct^2, it'd be something like:

Asin(x)+Bcos(x)+Ccos^2(x)+Dsin^2(x)

i'm not 100% sure, but i'd try something like that and see if it works out. good luck!
 

HallsofIvy

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You can't assume a solution of the form cos2 x or sin2 x because sine or cosine squared are not of the form that gets as solutions to a homogeneous equation with constant coefficients. However, you CAN use a trigonometric identity. Since cos(a+ b)= cos(a)cos(b)- sin(a)sin(b), taking a= b= x, cos(2x)= cos2(x)- sin2(x). Replacing cos2(x) by 1- sin2(x), cos(2x)= 1- 2sin2(x) so sin2(x)= (1/2)(1- cos(2x)). Look for a particular solution of the form A+ Bsin(2x)+ Ccos(2x).
 
Thx for helping ! i did solve the Q:!!)
 

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