Assuming your variable of integration is t, you would do this:
Differentiating yields
Can you go from here?
iknow that i should differentiate by x
but i dont know if it the correct way.
i know that differentiation cancels integration but here i have intervals with variables on the integrals
and i dont know what is y(x)
why you dissregarded the zero in the lower interval when you differentiate the integral.
and why the integra of ty(t) turns into xy(x)-xy(x) and not xy(x)-xy(0)
?
about the going further ,i tried is it ok
turns into
turns to
after another differentiation
turns to
correct?
Because, when you plug in the zero, the result is a constant. The derivative of a constant is zero.why you disregarded the zero in the lower interval when you differentiate the integral
.and why the integra of ty(t) turns into xy(x)-xy(x) and not xy(x)-xy(0)
It didn't come from that. You have to use the product rule on
You have a sign error there, as well as a losing of the constant. It should be
[EDIT]: See below for a correction.
Ah, but what you're talking about is only evaluating the integral in the first place. The next step in our problem is to differentiate. So yes, you start with
by the fundamental theorem of calculus, part 2. But then you differentiate thus:
by the fundamental theorem of calculus, part 1.
I know it is a bit late but there is a different approch using the LaPlace transform.
The left hand side is the convolution of two functions so its transform is the product of the 2.
Taking the transform gives
Solving this for gives
and inverting the transform gives