Method of Undetermined Coefficients

[polarbears]

Homework Statement

y⁽⁴⁾-4y²=t²+e^t
Determine the general solution

Homework Equations

The Attempt at a Solution

So I worked it all out and got the correct answer. My question is that when we make the onsat that a particular solution will be in the form Y=At⁴+...how do we know it starts with a 4th degree? It doesn't make sense to me since the right hand side is a 2nd degree

 

[miqbal]

You get better at guessing particular solutions.

First you have to get the complementary solution. Once you have that assume that the y is something like At² + Be^t. You're fine up until now as long as your complementary solution doesn't have the same term in it. Otherwise you will have to multiply your particular solution by t.

You complementary solution will be of the form Ate^t + Be^t. Since the particular solution has the same term Be^t, you multiply once by t to get Bte^t, but that's the same form as Ate^t. So you have to multiply by t again. Resulting in the 4th degree.