Integral equation with a derivative of the function inside the integral

In summary, the given integral equation is solvable and can be solved using the convolution theorem and Laplace transforms. By setting t=0, the constant f(0) can be determined. The final solution is f(t) = 64te^{8t} + 64sin(8t).
  • #1
damoj
9
0
[tex] f(x) = 2\int_{0}^{t} sin(8u)f'(t-u) du + 8sin(8t) , t\geq 0 [/tex]
is this problem solvable? I've never seen an integral equation like this with an f'(t-u)

i tried to solve it us the convolution theorem and laplace transforms but ended up with

[tex] s^{2} F(s) + 64F(s)- 16(F(s) - f(0)) =64 [/tex]
and i haven't been given f(0)
 
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  • #2
This is just a hunch, but if you assume f(t) = A sin(at + theta), you will wind up with an equation in terms of sines and cosines. Smart money says that "a" would be 8. Then it's a matter of solving algebraic equations to get the phase and amplitude. Another hunch: It may be easier to convert everything to exponentials and work it from there.
 
  • #3
damoj said:
[tex] f(x) = 2\int_{0}^{t} sin(8u)f'(t-u) du + 8sin(8t) , t\geq 0 [/tex]
is this problem solvable? I've never seen an integral equation like this with an f'(t-u)
I think you want f(t) on the LHS, right?
i tried to solve it us the convolution theorem and laplace transforms but ended up with

[tex] s^{2} F(s) + 64F(s)- 16(F(s) - f(0)) =64 [/tex]
and i haven't been given f(0)
What do you get if you set t=0 in the original equation?

I didn't bother trying to reproduce what you did, but your approach sounds fine. If you can't figure it out, show your work and we'll be able to provide more guidance.
 
  • #4
damoj said:
[tex] f(x) = 2\int_{0}^{t} sin(8u)f'(t-u) du + 8sin(8t) , t\geq 0 [/tex]
is this problem solvable? I've never seen an integral equation like this with an f'(t-u)

i tried to solve it us the convolution theorem and laplace transforms but ended up with

[tex] s^{2} F(s) + 64F(s)- 16(F(s) - f(0)) =64 [/tex]
and i haven't been given f(0)

You could always set f(0) = c (an unspecified constant) and get a solution in terms of t and c. However, if looks like you CAN get f(0) from the original integral equation, at least if f' does not have a singularity at zero.

RGV
 
  • #5
thanks for the replies

yeah it was meant to be f(t) on the LHS

Im going to have another look at it and get back to you guys.
 
  • #6
so i think i solved it, thanks guys.
heres the solution in case anyone has a similar one in the future.

[tex]f(t)=2∫t0sin(8u)f′(t−u)du+8sin(8t),t≥0[/tex]

set t=0
then f(0)= 0

the convolution gives us

[tex](sF(s) - f(0)) \cdot \frac{8}{s^{2}+64} [/tex]

and the rest is just algebra
[tex]f(t) = 64te^{8t}[/tex]
 

1. What is an integral equation with a derivative of the function inside the integral?

An integral equation with a derivative of the function inside the integral is an equation that involves both an integral and a derivative, where the function being differentiated is also the one being integrated.

2. What is the purpose of an integral equation with a derivative of the function inside the integral?

The purpose of an integral equation with a derivative of the function inside the integral is to express a relationship between a function and its derivative in terms of an integral. This can be useful in solving differential equations or finding solutions to other mathematical problems.

3. How is an integral equation with a derivative of the function inside the integral solved?

An integral equation with a derivative of the function inside the integral can be solved by using various techniques such as integration by parts, substitution, or other methods of integration. The specific method used will depend on the structure of the equation.

4. Are there any applications of integral equations with a derivative of the function inside the integral in real-world problems?

Yes, there are various applications of integral equations with a derivative of the function inside the integral in real-world problems. For example, they are commonly used in physics and engineering to model systems and analyze their behavior over time.

5. Can integral equations with a derivative of the function inside the integral be solved analytically?

In some cases, integral equations with a derivative of the function inside the integral can be solved analytically, meaning a closed-form solution can be found. However, in many cases, numerical methods may be necessary to approximate the solution due to the complexity of the equation.

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