Question

l lbout 0 for a general solution to the given differential equation u, y(0) = 0, V,(0) = 1 . Your answer should include a grneral formula for the ncients. (Find a recursive relation. If possible find Vi and 1,2). 3: Chebyshev's equation i(y + p'y-0, where p is a constant. Find two linearly Independert series solutions yi and ya. (Hint: find the series solution to the differential equation at z-0 to factor ao and ai as we discussed during the class.) 4: Hermite's equation is y" - 2ry +2py - 0, where p is a constant. Find two linearly independent series solutions V1 and V2. (Hint: find the series solution to the differential equation at z = 0, then try to factor ao and ai as we discussed during the class.) Notice that the above equations have the following property: when multiplied by suitable constants, the solutions are called Chebyshev polynomials or Hermite's in the study of the above differential equations in mathematical physics. polynomials. These polynomials arise naturally 5: Airy's equation is y"+y 0. The solutions are called Airy functions, which have applications to the theory of diffraction. Find two linearly independent series solutions n and y2. (Hint: find the series solution to the differential equation at x = 0; then try to factor ao and al as we discussed during the class.) Can you use the result you just found to write down the general solution of 't r-0 without calculation?
problem 5

Answer 1

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