Sec 5.3 – Undetermined Coefficients The Method Of Educated Guesses

We have seen how to guess at for some f’s Here is our setting If f(x) is a polynomial of degree n, we let And solve for A, B, … If f(x) is of the form , let And solve for A And finally if , let And solve for A, B

Just to recall The that we get is a solution to the original equation all by itself If we want a general solution or have initial conditions, we need to bring in

We can extend this a little bit If yp is a duplicate of one of the terms in the solution of the complementary homogeneous equation, we have to tweak it a little If f(x) is a (still manageable) combination of the forms we already have dealt with, we can get a yp

If yp duplicates a solution of complementary Has roots r = 1, r = 4 Therefore the fundamental set that goes with the complementary homogeneous equation is From last time, our guess solution for yp would be Since this is a linear combination of y1, y2, it will give 0 when we substitute it into the left hand side of the original, no matter what A is.

If yp duplicates a solution of complementary In keeping with our trick from earlier in Chapter 5, to rectify this duplication, we multiply the usual guess solution from last time by x: Solve for A by substituting into the original equation Then the general solution becomes

And the next step… Because the characteristic equation factors as (r+1)2 = 0, the fundamental set is So if we tried the trick from the preceding page, we would multiply in and x to get But that’s also a solution to homogeneous equation. Thus, to get a linearly independent yp, we multiply in another x Solve for A by substituting into the original equation. And the general solution becomes

Mixed force functions f(x) We will begin with a trick similar to that in Chapter 1, see where it takes us Assume that the form of yp is We will now substitute that into the original equation, set them combined terms = f(x) yp, and see that happens

If exp(x) is a duplicate: