Hint: Try calculating the derivatives of y(x) and then show algebraically that the relation exists (if it does exist).
I was given a Volterra integral equation y(x)=1/2*x^2+integral(0--->x) [t(t-x)y(t)]dt to solve using iteration. I have no idea how and where to start...
The full problem goes as follows:
Show that the solution y(x) of y''+xy=1, y(0)=y'(0)=1 also satisfies the integral equation (above). Use iteration to solve the integral equation, and compare with the series solution of the differential equation.
The series solution (done with taylor) equals: 1+x+1/2*x^2-1/2*x^3-1/6*x^4-.....
when you turn the differential equation to an integral equation it looks the same except it has additional 1+x that come from the initial conditions. So basically, to compare them we need different initial conditions i.e y(0)=y'(0)=0 and not 1. However I've asked my professor, and he said there is no such problem and that the algebra was wrong. I can't find the error...
Please help me figure this out.
Thanks to anyone that tries!