Proof that solution of integral equation satisfy the differential equation

[andkc]

Ηey!
I want your help to solve this:
Prove that,if f is the solution of integral equation
y(t)=exp(i*t)+a* integral(from t to inf) [sin(t-s)y(s)s^(-2)ds than f satisfy the differential equation y''+(1+at^-2)y=0
Please,if you could just give me a clue,I tried different ways but none of them was correct

 

[HallsofIvy]

What "different ways" did you try? You should always include your attempts so that we can see what you do or do not understand about the problem. In general one shows that a given function satisfies a differential equation by differentiating the function and putting its derivatives into the equation. Here, the given function is \(\displaystyle y(t)= e^{it}+ a\int_t^\infty sin(t-s)y(s)s^{-2}ds\).


The derivative of \(\displaystyle \int_t^A f(s, t) ds\), with respect to t, is \(\displaystyle -f(t, t)+ \int_t^A \frac{\partial f}{\partial t} ds\). What are the first and second derivatives of y(t)?

 

[Deleted member 4993]

Ηey!
I want your help to solve this:
Prove that,if f is the solution of integral equation
y(t)=exp(i*t)+a* integral(from t to inf) [sin(t-s)y(s)s^(-2)ds than f satisfy the differential equation y''+(1+at^-2)y=0
Please,if you could just give me a clue,I tried different ways but none of them was correct

.

Can you please share some work with us - at least list some of the ways you tried and where did you get stuck?