ODE ( 2nd order nonhomogeneous equation)

[soonsoon88]

Homework Statement

By using the method of undetermined coefficients,find the particular solution of
y''+y'+y=(sin x)^2

Homework 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

The Attempt at a Solution

 

[jimmypoopins]

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]

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).

 

[soonsoon88]

Thx for helping ! i did solve the Q:!)