Math 217 - November 19, 2010 Series Solutions 1. Find the recursion - - PowerPoint PPT Presentation

Math 217 - November 19, 2010

◮ Series Solutions

• 1. Find the recursion relation for the equation

x3y′ = 2y

• 2. Find the recursion relation for the equation

y′′ + y = x

• 3. Find the recursion relation for the equation

y′′ − 2y′ + y = 0

• 1. Find the recursion relation for the equation

x3y′ = 2y Solution: c0 = c1 = c2 = 0, cn+2 = ncn/2 and therefore cn = 0 for all n

• 2. Find the recursion relation for the equation

y′′ + y = x Solution: c2 = −c0/2, c3 = −(c1 − 1)/6, cn+2 = −cn/((n + 1)(n + 2)) for n > 1.

• 3. Find the recursion relation for the equation

y′′ − 2y′ + y = 0 Solution: cn+1 = 2ncn−cn−1

n(n+1)

• 4. (fun problem) Determine the sum

1 1 − 1 3 + 1 5 − 1 7 + · · ·

• 4. (fun problem) Determine the sum

1 1 − 1 3 + 1 5 − 1 7 + · · · Solution: π/4, this is the series for arctan x with x = 1.

Lecture Problems

• 5. For the recursive relation, find the series.

c0 = c1 = c2 = 0, cn+2 = ncn 2

• 6. For the recursive relation, find the series.

c2 = −c0 2 , c3 = −c1 − 1 6 , cn+2 = − cn (n + 1)(n + 2) for n ≥ 2

• 7. Use initial conditions y(0) = 0 and y′(0) = 1, use the recursive

relation to find the series. cn+1 = 2ncn − cn−1 n(n + 1)

Lecture Problems

• 5. For the recursive relation, find the series.

c0 = c1 = c2 = 0, cn+2 = ncn 2 Solution: y = 0

• 6. For the recursive relation, find the series.

c2 = −c0 2 , c3 = −c1 − 1 6 , cn+2 = − cn (n + 1)(n + 2) for n ≥ 2 y = x + c0 cos x + (c1 − 1) sin x

• 7. Use initial conditions y(0) = 0 and y′(0) = 1, use the recursive

relation to find the series. cn+1 = 2ncn − cn−1 n(n + 1) Solution: cn =

1 (n−1)!, y = xex