Worksheet: Power Series Solutions of Differential equations

In this worksheet, we will practice finding solution to linear differential equations with variable coefficients about an ordinary point using power series.

Q1:

Which of the following statements is true with regard to a power series solution to the ordinary differential equation ( π‘₯ βˆ’ 1 ) 𝑦 β€² β€² + 1 π‘₯ 𝑦 β€² βˆ’ 2 𝑦 = 0 ?

  • AInsufficient information has been provided to distinguish whether π‘₯ = 0 and 1 are ordinary points, regular singularities, or irregular singularities.
  • B π‘₯ = 0 and 1 are regular singularities.
  • C π‘₯ = 0 and 1 are ordinary points.
  • D π‘₯ = 0 and 1 are irregular singularities.

Q2:

Give the first four terms of the power series solution for the following differential equation: 𝑦 β€² β€² + ( π‘₯ ) 𝑦 β€² + 𝑒 𝑦 = 0 s i n  .

  • A 𝑦 = 𝑐 ο€Ό 1 βˆ’ 1 2 π‘₯ βˆ’ 1 6 π‘₯ + 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―  + 𝑐 ο€Ό π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―    οŠͺ      οŠͺ
  • B 𝑦 = 𝑐 ο€Ό 1 βˆ’ 1 2 π‘₯ βˆ’ 1 6 π‘₯ + 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―  + 𝑐 ο€Ό π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―              
  • C 𝑦 = 𝑐 ο€Ό 1 βˆ’ 1 2 π‘₯ βˆ’ 1 6 π‘₯ + 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―  + 𝑐 ο€Ό π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―     οŠͺ    οŠͺ 
  • D 𝑦 = 𝑐 ο€Ό 1 βˆ’ 1 2 π‘₯ + 1 6 π‘₯ + 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―  + 𝑐 ο€Ό π‘₯ + 1 3 π‘₯ + 1 1 2 π‘₯ + 1 2 0 π‘₯ + β‹―     οŠͺ    οŠͺ 

Q3:

Find the series solution for the following ordinary differential equation using the Frobenius method: π‘₯ 𝑦 + π‘₯ ο€Ό π‘₯ + 1 2  𝑦 βˆ’ ο€Ό π‘₯ + 1 2  𝑦 = 0 .     

  • A 𝑦 = 𝑐 π‘₯ ο€Ό 1 βˆ’ 2 5 π‘₯ + 9 3 5 π‘₯ βˆ’ 8 2 9 4 5 π‘₯ + 5 7 1 2 0 7 9 0 π‘₯ βˆ’ β‹―  + 𝑐 π‘₯ ο€Ό 1 βˆ’ π‘₯ + 3 2 π‘₯ βˆ’ 1 3 1 8 π‘₯ + 1 1 9 3 6 0 π‘₯ βˆ’ β‹―     οŠͺ    οŠͺ  
  • B 𝑦 = 𝑐 π‘₯ ο€Ό 1 βˆ’ 2 5 π‘₯ + 9 3 5 π‘₯ βˆ’ 8 2 9 4 5 π‘₯ + 5 7 1 2 0 7 9 0 π‘₯ βˆ’ β‹―  + 𝑐 π‘₯ ο€Ό 1 βˆ’ π‘₯ + 3 2 π‘₯ βˆ’ 1 3 1 8 π‘₯ + 1 1 9 3 6 0 π‘₯ βˆ’ β‹―     οŠͺ     οŠͺ  
  • C 𝑦 = 𝑐 π‘₯ ο€Ό 1 βˆ’ 2 5 π‘₯ βˆ’ 9 3 5 π‘₯ + 8 2 9 4 5 π‘₯ βˆ’ 5 7 1 2 0 7 9 0 π‘₯ + β‹―  + 𝑐 π‘₯ ο€Ό 1 + π‘₯ βˆ’ 3 2 π‘₯ + 1 3 1 8 π‘₯ βˆ’ 1 1 9 3 6 0 π‘₯ + β‹―     οŠͺ    οŠͺ  
  • D 𝑦 = 𝑐 π‘₯ ο€Ό 1 βˆ’ 2 5 π‘₯ βˆ’ 9 3 5 π‘₯ + 8 2 9 4 5 π‘₯ βˆ’ 5 7 1 2 0 7 9 0 π‘₯ + β‹―  + 𝑐 π‘₯ ο€Ό 1 + π‘₯ βˆ’ 3 2 π‘₯ + 1 3 1 8 π‘₯ βˆ’ 1 1 9 3 6 0 π‘₯ + β‹―     οŠͺ     οŠͺ  

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