 #1
0kelvin
 50
 5
 Homework Statement:

Let f be a continuous function in a interval I containing the origin and let
##y = y(x) = \int_0^x sin(x  t)f(t) dt##
Prove that ##y'' + y = f(x)## and ##y(0) = y'(0) = 0## for all x ##\in I##
 Relevant Equations:
 ...
I know how to solve ##\frac{d}{dx} \int_0^{x^2} sin(t^2) dt## and from the statement I got that f(0) = 0 because f contains the origin and is continuous.
I tried y'(x) = sin(x  x)f(x)  sin(x  0)f(0) but that doesn't seem to look good.
I tried y'(x) = sin(x  x)f(x)  sin(x  0)f(0) but that doesn't seem to look good.