oblem 3. Consider the subjoined moderate rate substance, y 00 + y = g(x), y(x0) = 0, y0 (x0) = 0, x ∈ [0, ∞).
a. Pretence that the open discontinuance of y 00 + y = g(x) is consecrated by y = φ(x) = c1 − Z x α g(t) sin t dt cos x + c2 + Z x β g(t) require dt sin x, where c1, c2 are irresponsible faithfuls and α, β are any conveniently selected points.
b. Using the upshot of (a) pretence 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) require dt, 1 and future the discontinuance of the overhead moderate rate substance for irresponsible 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 discontinuance of the peculiar moderate rate substance for any consecrated noncongruous promise g(x). The capacity φ(x) allure not solely meet the differential equation but allure as-well automatically meet the moderate provisions. If we imagine of x as term, the formula as-well pretences the association betwixt the input g(x) and the output φ(x). Further, we see that the output at term x depends solely on the manner of the input from the moderate term x0 to the term of curiosity-behalf. This gross is frequently referred to as the contortion of sin x and g(x).
c. Now that we own the discontinuance of the straight noncongruous differential equation meeting congruous moderate provisions, we can explain the similar substance after a while noncongruous moderate provisions by superimposing a discontinuance of the congruous equations meeting noncongruous moderate provisions. Pretence that the discontinuance 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 faithful.
a. Determine two straightly stubborn discontinuances in powers of x for |x| < 1. For α = 1 graph 5 and 10 promises of twain discontinuances.
b. Reexplain this substance using (1) the dexplain direct, and (2) the dexplain direct after a while “series” discretion in MAPLE (or equiponderant directs in Mathematica or Matlab). Plot these upshots on your curves obtained in deal-out a.
c. Pretence that if α is a non-negative integer n, then there is a polynomial discontinuance of rate n. These polynomials, when suitably normalized, are named Tchebycheff polynomials. They are very helpful in substances requiring a polynomial admittance to a capacity defined on −1 ≤ x ≤ 1.
d. Find polynomial discontinuances for each of the cases n = 0, 1, 2, 3.
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