- #1
erba
- 10
- 1
Homework Statement
Solve the integral
[itex]
y(t) + \int_0^t (t-u)y(u) \, du = 3sin(2t)
[/itex]
Homework Equations
The Attempt at a Solution
Rewrite the equation:
[itex]
y(t) = 3sin(2t) - \int_0^t (t-u)y(u) \, du
[/itex]
I assume the integral to be the convolution:
[itex]
f(t) * y(t) = t * y(t)
[/itex]
as
[itex]
f(t-u) = f(t) = t
[/itex]
when
[itex]
u = 0
[/itex]
Laplace transform both sides, and after simplification get:
[itex]
y(s) = \frac{6s^2}{(s^2 + 4)(s^2 + 1)}
[/itex]
Apply partial fraction decomposition:
[itex]
y(s) = \frac{A}{s^2 + 4} + \frac{B}{s^2 + 1}
[/itex]
and I find that:
[itex]
A = 8
[/itex]
[itex]
B = -2
[/itex]
Thus,
[itex]
y(s) = \frac{8}{s^2 + 4} - \frac{2}{s^2 + 1}
[/itex]
Inverse Laplace transform then gives:
[itex]
y(t) = 4sin(2t) - 2sin(t)
[/itex]
But then I want to double-check myself by substituting y(t) into the original equation, that we were supposed to solve. I.e.:
[itex]
y(t) + \int_0^b (t-u)y(u) \, du = 3sin(2t)
[/itex]
[itex]
4sin(2t) - 2sin(t) + (4sin(2t) - 2sin(t)) \int_0^t (t-u) \, du = 3sin(2t)
[/itex]
[itex]
4sin(2t) - 2sin(t) + \frac{t^2}{2}(4sin(2t) - 2sin(t)) \neq 3sin(2t)
[/itex]
But LHS is not equal to RHS. Where is my mistake?
My intuitive feeling is that the error lies in my "partial fraction decomposition", as all the other processes are quite straight forward.
Thank you very much!