- #1

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Ok, I have a quick question (hopefully). I'm trying to use the method of undetermined coefficients, and I keep getting stuck at one specific spot in the method. I'm not exactly sure what I'm doing. Let me try and explain:

The problem is given as:

[tex]2y''+3y'+y=t^2+3*\sin t[/tex]

Which leads to:

[tex]y=y_p+y_c|y_c=c_1 e^{\frac{-t}{2}} + c_2 e^{-t}[/tex]

Now, I'm sure that the [tex]y_c[/tex] portion is correct. It is the [tex]y_p[/tex] part that I get confused on.

I'll go through my steps, so you can see what I am doing right/wrong.

So, we first split [tex]y_p[/tex] as follows:

[tex]y_p=y_{p1}+y_{p2} [/tex]

Where:

(*1) - [tex] y_{p1} [/tex] satisfies [tex]2y_{p1}''+3y_{p1}'+y_{p1}=t^2[/tex]

(*2) - [tex] y_{p2} [/tex] satisfies [tex]2y_{p2}''+3y_{p2}'+y_{p2}=3\sin t [/tex]

For the [tex] y_{p1} [/tex] portion:

[tex] y_{p1} = At^2+Bt+C [/tex]

[tex] y_{p1}'' = 2At+B [/tex]

[tex] y_{p1}'' = 2A [/tex]

Plugging into (*1) yields:

[tex] 2[2A]+3[2At+B]+[At^2+Bt+C] = t^2 [/tex]

[tex] [A]t^2 + [6A+B]t^1 +[4A+3B+C]t^0 = t^2 [/tex]

Now this is where I get confused.

I'm supposed to factor and arrange the terms, and setup a system of equations?

So, maybe something like this?

[tex] t^2: A = \lambda_1 [/tex]

[tex] t^1: 6A + B = \lambda_2 [/tex]

[tex] t^0: 4A+3B+C= \lambda_3 [/tex]

Now, how do I know what [tex]\lambda_n [/tex] are? The book, seems to magically find a number for them, but I'm NOT sure where those numbers are coming from. So if someone could explain this step, I would be very thankful. I think once I understand this step that I will be able to carry on with the other problems and do the [tex] y_{p2} [/tex] portion of this problem also.

Thanks in advance :)