Problem 3. Consider the following initial value problem, y 00 + y = g(x), y(x0) = 0, y0 (x0) = 0, x ∈ [0, ∞).
a. Show that the general solution of y 00 + y = g(x) is given by y = φ(x) = c1 − Z x α g(t) sin t dt cos x + c2 + Z x β g(t) cost dt sin x, where c1, c2 are arbitrary constants and α, β are any conveniently chosen points.
b. Using the result of (a) show that y(x0) = 0 and y 0 (x0) = 0 if, c1 = Z x0 α g(t) sin t dt, c2 = − Z x0 β g(t) cost dt, 1 and hence the solution of the above initial value problem for arbitrary g(x) is, y = φ(x) = Z x x0 g(t) sin(x − t) dt. Notice that this equation gives a formula for computing the solution of the original initial value problem for any given nonhomogeneous term g(x). The function φ(x) will not only satisfy the differential equation but will also automatically satisfy the initial conditions. If we think of x as time, the formula also shows the relation between the input g(x) and the output φ(x). Further, we see that the output at time x depends only on the behavior of the input from the initial time x0 to the time of interest. This integral is often referred to as the convolution of sin x and g(x).
c. Now that we have the solution of the linear nonhomogeneous differential equation satisfying homogeneous initial conditions, we can solve the same problem with nonhomogeneous initial conditions by superimposing a solution of the homogeneous equations satisfying nonhomogeneous initial conditions. Show that the solution of y 00 + y = g(x), y(x0 = 0) = y0, y0 (x0 = 0) = y 0 0 , is y = φ(x) = Z x x0 g(t) sin(x − t) dt + y0 cos x + y 0 0 sin x.
Problem 4. The Tchebycheff (1821-1894) differential equation is (1 − x 2 )y 00 − xy0 + α 2 y = 0, α constant.
a. Determine two linearly independent solutions in powers of x for |x| < 1. For α = 1 graph 5 and 10 terms of both solutions.
b. Resolve this problem using (1) the dsolve command, and (2) the dsolve command with “series” option in MAPLE (or equivalent commands in Mathematica or Matlab). Plot these results on your curves obtained in part a.
c. Show that if α is a non-negative integer n, then there is a polynomial solution of degree n. These polynomials, when properly normalized, are called Tchebycheff polynomials. They are very useful in problems requiring a polynomial approximation to a function defined on −1 ≤ x ≤ 1.
d. Find polynomial solutions for each of the cases n = 0, 1, 2, 3.