Question B5 (10 marks) (a) Prove the following identity for the digamma function or psi function: (1-2)=(2)+cot where the digamma function is defined as (4 marks) 1"(2) y(z) = in (2)=(x) Hint: The reflection formula derived in Question 4 for the gamma function may be useful: г(1-2) (2)=- sin (b) Find the solution of the following non-constant coefficient ODE by the Laplace transform: +7ry 0, y(0)-3, y(0)=0 (6 marks) dt Hints: Ly(t))-Y(s)=ey(1)d; y'=; d" y(t) [0]=(s)-(0)-(0)-..--(0) dr" Jo(at)={ a=' L[ry]=(-1) (s)=(-1)*y()(s) ds" sinh vat, kh =cosh at, =cos at, I' s-a b e" cos bt, L esin br, (-a)²+b² r[o sin at, L (s-a)+b sinat,] sin at-at cos at (3-ar)sin at-3at cos at [[]]- 8a 2a End of Section B