Undetermined coefficients why doesn't it work?

[1MileCrash]

Homework Statement

y''+y=3sin2t+2tcos2t

Homework Equations

The Attempt at a Solution

I tried doing this problem by breaking it into 3 others. It doesn't quite work, and I want to know why.

First, I solved for the nonhomog part of:

y''+y=3sin2t

Which is Y1 = -sin2t

Then, I did the same with

y''+y = 2t

which is Y2 = 2t

Lastly, I did

y''+y=cos2t

Which is Y3 = -(1/3)cos2t



Combining these with the homog. parts gives me:

y = c1cost + c2sint - (2/3)tcos2t -sin2t


But, this is not quite right, as the -sin2t should be -(1/9)sin2t.

What is incorrect about breaking the DE up and putting it back together in this way?

Thank you!

 

[vela]

You don't split up products in the forcing function when relying on superposition. Your first solution is fine, but you need to solve
$$y''+y = 2t\cos 2t$$ to get the second solution, which you can then add it to the first to find the particular solution.

 

[1MileCrash]

You don't split up products in the forcing function when relying on superposition. Your first solution is fine, but you need to solve
$$y''+y = 2t\cos 2t$$ to get the second solution, which you can then add it to the first to find the particular solution.

I really thought that was the case before starting.

Is this the case because:

d/dt(f(t) + g(t)) = f'(t) + g'(t)

but
d/dt(f(t)g(t)) =/= f'(t)g'(t)

?

 

[vela]

Yeah, pretty much.

 

[1MileCrash]

This is kind of interesting.

Thanks, now I can solve these problems.

But, I wonder, with these terms I have found before, is there a way to construct the answer? Applying the product rule to the second two terms and adding that to the first doesn't work. I'm trying to think about what's going on and how those terms relate to the correct answer, since it's close. I tend to do that.

 

[haruspex]

This is kind of interesting.

Thanks, now I can solve these problems.

But, I wonder, with these terms I have found before, is there a way to construct the answer? Applying the product rule to the second two terms and adding that to the first doesn't work. I'm trying to think about what's going on and how those terms relate to the correct answer, since it's close. I tend to do that.

Suppose L(f(t)) = u(t) and L(g(t)) = v(t), where L(h) = h"+h.
L(fg) = f"g + 2f'g' + fg" + fg
uv = f"g + fg" + fg + f"g"
L(fg) - uv = 2f'g' - f"g"
In this example, f" = k g' etc., so it almost worked.