MEGN 502

Advanced Engineering Analysis

Assignment No. 2

1 September 2015

1. Determine the solution for the following ordinary differential equation

d2y

dx2+ x

dy

dx− 4y = 0

using the power series,

y(x) =∞∑k=0

Akxk

Provide a recurrence relation for the Ak.

2. Determine the solution for the following ordinary differential equation(1− 1

2x2)d2y

dx2+ x

dy

dx− y = 0

using the power series,

y(x) =∞∑k=0

Akxk

Provide a recurrence relation for the Ak.

3. Locate and classify the singular points of Problems 1 and 2.

4. Locate and classify the singluar points of the following differential equation

d2y

dx2+ log x

dy

dx+ xy = 0, x ≥ 0

1

5. Use the Method of Frobenius to obtain the general solution of the followingdifferential equation valid near x=0:

x(1− x)d2y

dx2− 2

dy

dx+ 2y = 0

6. The differential equation

(1− x2)d2y

dx2− cx

dy

dx+ n(n + c− 1)y = 0

is a specialization of Jacobi’s Equation. Show that if n is nonzero or apositive integer it possesses a polynomial solution of the form

un(x) = 1− n(n + c− 1)

2!x2 +

{n(n− 2)}{(n + c− 1)(n + c + 1)}4!

x4 + · · ·

when n is even, and of the form

vn(x) = x− (n− 1)(n + 2)

3!x3+{(n− 1)(n− 3)}{(n + c)(n + c + 2)}

5!x5−· · ·

when n is odd.

2