GET 311 Problem Set 1 (due for submission by 3 pm on Friday, 17/03/2023)
1. Find the first five terms in the power series solution of the following initial value
problem:
𝒅𝒚
+ (𝒙 − 𝟐𝒙𝟐 )𝒚 = 𝟏, 𝒘𝒊𝒕𝒉 𝒚(𝟎) = 𝟏
𝒅𝒙
2. Find the terms up to x4 in the power series solution of the following initial value
problem:
𝒚′′ + (𝟏 + 𝟐𝒙)𝒚′ + 𝒙𝒚 = 𝟎, with y(2) = 1 and 𝒚′ (𝟐) = 𝟎
3. Derive the recurrence relation and find two linearly independent solutions of the
following Legendre differential equation:
(𝟏 − 𝒙𝟐 )𝒚′′ − 𝟐𝒙𝒚′ + 𝟔𝒚 = 𝟎
4. Find two linearly independent solutions of the following ODE for x>0, and
determine at least the first four leading terms in the second solution y(2).
𝟗𝒙𝟐 𝒚′′ − 𝟔𝒙𝒚′ + 𝟐𝒚 = 𝟎
5. Use the Frobenius method to find two linearly independent solutions of the following
equation about the origin.
𝟒𝒙𝒚′′ + 𝟐𝒚′ + 𝒚 = 𝟎