Derive the solution of the ordinary differential equation
d2y/dx2 =f(x), x>0, y(x)=0, dy/dx (0)=0,
in form y(x)= integral from 0 to x [(x-t)f(t)dt]
Are you asked to actually prove the formula, or just show this is the solution? Because the latter is easy: Differentiate twice (under the integral sign) to get y"=f, and since y also gives y(0)=0, y'(0)=0, from the uniqueness theorem this is the only solution!
Yes, I know it's cheating, but it's a way. Now if you are asked to derive the formula, write
and integrating by parts,
since y''=f. Now one integration of the differential equation gives us , and substitute into (1) to get
The function must also satisfy the initial conditions.To show this just put...