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1. Find the indical roots of the following differential equation
x^2 y" + 6xy' + (6 - x^2)y = 0 (10 points)
2. Write down the Legendre's differential equation and Bessel's differential
equation of order n. (10 points)
∞ (-1)^n x
3. From Jυ(x) = Σ ---------------- (---)^2n+υ , show that
n=0 n！Γ(n+υ+1) 2
1 J 2
(a) (x^υ Jυ)' = x^υ Jυ-1 (b) Γ(---) = √π (c) 1/2 = √(---) sinx
2 πx
(20 points)
1
4. Show that ∫ Pn(x)Pm(x) dx = 0 for n ≠ m (20 points)
-1
5. Find the inverse Laplace transform of the following functions :
s e^-2s s
(1) ㏑(-----) (2) -------- (3) ----------- by convolution thm. (20 points)
1+s s - 4 (s^2 + 1)^2
6. Find the Laplace transform of the following functions (20 points)
(1) e^-t sin2t (2) sin^2 t (3) t^2 u(t-1)
7. Use the Laplace transform to solve the following problem
y" + 2y' + 2y = δ(t-1) ； y(0) = y'(0) = 0 (20 points)
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※ 編輯: GhostWall 來自: 220.139.74.11 (01/24 16:57)
※ 編輯: GhostWall 來自: 220.139.74.11 (01/24 17:00)