Problem 2 a) The fundamental solutions to the Legendre equation are: Y₁(x) = 1 - y₂(x) = x (n-1)(n + 2) 3! Show that when n = 0 Y₁(x) = 1 1- n(n+1) 2! y2(x) = x + 1 3 +. 5 2 x² + + (n − 2)n(n + 1)(n + 3) 4! + (n-3)(n T X 4 - +... 1)(n + 2)(n + 4) 5! x5 3 b) Solve the Legendre equation using separation of variables for n=0. Legendre equation: (1-x²)y" - 2xy' + n(n + 1)y = 0 +.. (n constant)

Q: Consider the initial value problem y-7 5t² 2 Existence and uniqueness is guaranteed at all points…

A:

Q: Problem 8. Find the number of zeroes (with multipticity) of the potynomtai 2 +z+1 n the annulus 1<…

A:

Q: (a)(√2-i)-i(1 – √√2i) = −2i; (Ⓒ) (3, 1)(3, −1) (3, 1) = (2, 1). (c 5' 10 Show that (a) Re(iz) = - Im…

A: Few Complex number expressions and equations are given.

Q: 1) Prove the identity 2n (²x) = ² (²) + ²n (2) 2 2n 3 3

A:

Q: 9. The general solution of x' = -6 -4 x is: 2 | ++(-:)| (a) x = Cịe-2t + Cze¬2t (b) x = Cye" () +…

A:

Q: Problem 6.37

A: Since f(x) is of degree 5, then f(x)-1 is also of degree 5. And given that f(x)-1 is divisible by…

Q: Theorem 5.6 For any integers a and b not both zero, the Diophantine equation ax + by gcf(a, b) = has…

A:

Q: Show that r n2n-1 r=0

A: Want to  prove:    ∑r=0n rnr=n2n-1

Q: 3. Algebraically solve the equation 2(C₂) ==(n+1²3) 5

A: Given 2(C2n)=15P3n+1

Q: Divide P(x) = x° + 2x2 - x – 5 by x - 3. Write your answer in the form R(x) Q(x) + D(x) 37 O22 + 5x…

A:

Q: 2. (a) Show that 25 – 32 = (z – 2)(16 + 8z + 4z² + 2z³ + z4). (b) Find the roots of 16 + 8z + 4z2 +…

A: