Thread: Solving for A,B w/ Method of Undetermined Coefficients

In trying to find the particular solution to a non-homogeneous third-order ODE, I used the method of undetermined coefficients but it seems to be the method of undeterminable coefficients.

I have

y^(3) - y^(1) = 2sint

So I assumed

Y(t) = Acost + Bsint

Differentiating and plugging in, I get

(Asint - Bcost)^3 + Asint - Bcost = 2sint

In solving these problems before, I would follow the idea, "There is a 0cost on the right, so the collected coefficients of the cost on the left must be zero." Applying this to the (sint)^3, (sint)^2(cost), (sint)(cost)^2, (cost)^3, sint, and cost, I would think that

A^3 = 0
-(A^2)B = 0
A(B^2) = 0
-B^3 = 0
A = 2
-B = 0

However, A cannot both be two and its cube be zero. Also, the back of the book gives Y(t) as cost, for which, if Y(t) = Acost + Bsint, A would be one.

I know I am solving for the coefficients incorrectly, but I don't know why my method is wrong here (since I know it has worked in other situations). How can I solve for these coefficients? Is my assumption of what Y(t) is wrong? Should it be something other than Acost + Bsint?

Thank you!

Originally Posted by Oijl

In trying to find the particular solution to a non-homogeneous third-order ODE, I used the method of undetermined coefficients but it seems to be the method of undeterminable coefficients.

I have

y^(3) - y^(1) = 2sint

So I assumed

Y(t) = Acost + Bsint

Differentiating and plugging in, I get

(Asint - Bcost)^3 + Asint - Bcost = 2sint

In solving these problems before, I would follow the idea, "There is a 0cost on the right, so the collected coefficients of the cost on the left must be zero." Applying this to the (sint)^3, (sint)^2(cost), (sint)(cost)^2, (cost)^3, sint, and cost, I would think that

A^3 = 0
-(A^2)B = 0
A(B^2) = 0
-B^3 = 0
A = 2
-B = 0

However, A cannot both be two and its cube be zero. Also, the back of the book gives Y(t) as cost, for which, if Y(t) = Acost + Bsint, A would be one.

I know I am solving for the coefficients incorrectly, but I don't know why my method is wrong here (since I know it has worked in other situations). How can I solve for these coefficients? Is my assumption of what Y(t) is wrong? Should it be something other than Acost + Bsint?

Thank you!

I guess I 'm a little confused over your notation...

The way I read your problem is this



Is this is the case your auxillary equation is



so

Now \





Plugging into the original we get





This tells us that A=0 and B=1

So

Doh! Of course, thanks, just needed another pair of eyes. I had plugged the derivatives of the particular solution into the auxiliary equation, somehow. I need better bookkeeping while doing problems, I guess. Thanks.