Hwk sheet 15 - MATH11007 SHEET 15: DIFFERENCE EQUATIONS Set...

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MATH11007 SHEET 15: DIFFERENCE EQUATIONS Set on Tuesday, April 21: Qs 1, 2 and 3. (1) Solve the following difference equations: (a) x n +2 - x n = 0 (b) x n +2 + 3 x n +1 - 7 4 x n = 9 (c) x n +2 - 3 x n +1 - 4 x n = 36 n , with x 1 = 1 and x 2 = 5 (d) x n +2 + 2 x n +1 + x n = 9 × 2 n (2) Consider a difference equation x n +2 + ax n +1 + b = 0. Write down a condition on a and b such that the characteristic equation, m 2 + am + b = 0, has a double root (i.e. only one solution). If ˜ m is the unique solution of the characteristic equation, show that x n = n ˜ m n is a solution of the difference equation. (3) For n N , define T n ( x ) := cos ( n arccos x ) . (a) Show that T 0 and T 1 are polynomials of degree 0 and 1 respectively. (b) Show that T n +1 ( x ) + T n - 1 ( x ) = 2 xT n ( x ) . Hence, prove by induction on n that T n is a polynomial of degree n . These polynomials are called the Chebyshev polynomials. (c) Find the roots of
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Hwk sheet 15 - MATH11007 SHEET 15: DIFFERENCE EQUATIONS Set...

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