MATH323 Differential Equation

Using separation of variables, write down the general solution to the initial value problemutt=uxx,u(0,t) = 0,ux(1,t) = 0,u(x,0) =f(x),ut(x,0) =g(x)The final answer should have the Fourier coefficients off(x) andg(x) in the answer.3.  Using the method of characteristics, solve the first order PDEut+ (x+t)ux= 0,  u(x,0) =f(x)In class,  we solved the problems with the specific initial conditionu(x,0) =f(x) =λx,  because anyfunction more complicated than this causes algebraic problems if you want an actual formula for theanswer.  THIS problem is such that you can write down the solution no matter whatf(x) is (because ithappens to be a LINEAR pde).4.  In class, we talked about the calculus of variations problem of finding the path of least time for light raysfor light travelling in a material where the index of refraction is given by the formulan(x,y) = 1/y2; itonly depends on the distance from thexaxis.  Also, note the index of refraction is theinverseof thespeed,  so as you get closer to theyaxis,  light travels more slowly.  The light paths turned out to becircles.  For this problem, I want you to find the path of least time with a different index of refraction