The "goal of the next step" is to solve for the constants A and B, right? That's what I hoped you'd tell me, because your teacher should have told you what you were trying to do. The goal of the whole problem is important, too, and we'll get to that, but it's not the immediate goal.

But you seem to have missed my point, so I'll say it a third time, with more context. In solving a differential

equation, you are trying to find a

**function**, y = f(x) that makes the differential equation true. This means you are not just talking about equations that are true for certain values of x, but rather about things that are true for ALL x. That's what a function is.

You assumed that f(x) = A cos(x) + B sin(x), where A and B are some as yet

**unknown constants**. Right now, you need to find their values. Keep in mind that they must be

**constants**. You're not solving for A as a function of x.

So you want -(B + 2A)sin(x) + (A - 2B)cos(x) = sin x to be true for ALL x. Simplify it to (A - 2B) cos(x) = sin x + (B + 2A) sin(x), and further to (A - 2B) cos(x) = (1 + B + 2A) sin(x). Don't expand it, as you did; that doesn't help. My point was that the only way a multiple of cos(x) can ALWAYS equal a multiple of sin(x), is if both multiples are ZERO. Think about that, until you see WHY it has to be true. This is, as I've said, a key idea here, and probably the one idea that is most preventing you from following what your teacher says.

Now, this means that

A - 2B = 0

1 + B + 2A = 0

Can you see that? When this is true, then the equation is 0 cos(x) = 0 sin(x), which is indeed true for all x.

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