Unformatted text preview: nous solution will all ways make the left side = 0). If you are not too sure what to try, look at the right side and think of functions that could derive to make that function. If the function is part of the homogenous solution, then place x powers as a multiplication in front of the term you had in mind (no more than x2 as otherwise you will not be able to reach the function after 2 derivatives).
Step 3: Substitute in your yp into your ODE and perform the derivatives. Simplify this as much as possible. Then identify the different terms and their coefficients. (You may have to factor here)
Step 4: Set the coefficients equal to each other to solve for the undetermined numbers.
Step 5: Write your solution as y=yh+yp Undetermined Coefficients for Sinusoidal
Say we had: ay ' 'by 'cy g ( x) g ( x) aCos (cx) bSin(cx)
Where: a,b,c here are some given numbers
Potential Candidates: g ( x) aCos (cx) bSin(cx)
If is not part of the homogenous solution, our candidate to try would be y p ACos (cx) BSin(cx) y p AxCos (cx) BxSin(cx)
Next we should try: unless it is part of the homogenous solution.
Note that you should not need to try anything else, as higher powers of x tend to give inconsistent systems. Undetermined Coefficients for Polynomials
Say we had: ay ' 'by 'cy g ( x) g ( x) a0 a1 x a2 x 2 ... an x n
Where: the a’s here are some given numbers
If is not part (any of the terms) of the g ( x) a0 a1 x a2 x 2 ... an x n
homogenous solution, our candidate to try would be y p A0 A1 x A2 x 2 ... An x n
Note that you should not need to try anything else, as higher powers of x will not be able to cancel out (derivatives always make it smaller). Undetermine...
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This note was uploaded on 02/07/2014 for the course MATH 1004 taught by Professor Mark during the Summer '00 term at Carleton CA.
- Summer '00
- Calculus, SmartPen Lectures, Lecture Notes, MATH1004, Kyle Harvey