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′′+1π‘₯π‘¦β€²βˆ’2𝑦=0?

  • Aπ‘₯=0 and 1 are ordinary points.
  • Bπ‘₯=0 and 1 are irregular singularities.
  • Cπ‘₯=0 and 1 are regular singularities.
  • DInsufficient information has been provided to distinguish whether π‘₯=0 and 1 are ordinary points, regular singularities, or irregular singularities.

Q2:

Give the first four terms of the power series solution for the following differential equation: 𝑦′′+(π‘₯)𝑦′+𝑒𝑦=0sin.

  • A𝑦=𝑐1βˆ’12π‘₯βˆ’16π‘₯+112π‘₯+120π‘₯+β‹―οˆ+𝑐π‘₯βˆ’13π‘₯βˆ’112π‘₯+120π‘₯+β‹―οˆοŠ¦οŠ¨οŠ¬οŠ§οŠ¨οŠ¨οŠ¦οŠ§οŠ©οŠ§οŠ¨οŠ¨οŠ¦
  • B𝑦=𝑐1βˆ’12π‘₯βˆ’16π‘₯+112π‘₯+120π‘₯+β‹―οˆ+𝑐π‘₯βˆ’13π‘₯βˆ’112π‘₯+120π‘₯+β‹―οˆοŠ¦οŠ©οŠͺοŠͺ
  • C𝑦=𝑐1βˆ’12π‘₯βˆ’16π‘₯+112π‘₯+120π‘₯+β‹―οˆ+𝑐π‘₯βˆ’13π‘₯βˆ’112π‘₯+120π‘₯+β‹―οˆοŠ¦οŠ¨οŠ©οŠͺοŠͺ
  • D𝑦=𝑐1βˆ’12π‘₯+16π‘₯+112π‘₯+120π‘₯+β‹―οˆ+𝑐π‘₯+13π‘₯+112π‘₯+120π‘₯+β‹―οˆοŠ¦οŠ¨οŠ©οŠͺοŠͺ

Q3:

Find the series solution for the following ordinary differential equation using the Frobenius method: π‘₯𝑦+π‘₯ο€Όπ‘₯+12οˆπ‘¦βˆ’ο€Όπ‘₯+12οˆπ‘¦=0.

  • A𝑦=𝑐π‘₯ο€Ό1βˆ’25π‘₯βˆ’935π‘₯+82945π‘₯βˆ’57120,790π‘₯+β‹―οˆ+𝑐π‘₯ο€Ό1+π‘₯βˆ’32π‘₯+1318π‘₯βˆ’119360π‘₯+β‹―οˆοŠ§οŠ¨οŠ©οŠͺοŠͺ
  • B𝑦=𝑐π‘₯ο€Ό1βˆ’25π‘₯βˆ’935π‘₯+82945π‘₯βˆ’57120,790π‘₯+β‹―οˆ+𝑐π‘₯ο€Ό1+π‘₯βˆ’32π‘₯+1318π‘₯βˆ’119360π‘₯+β‹―οˆοŠ§οŠ¨οŠ©οŠͺοŠͺ
  • C𝑦=𝑐π‘₯ο€Ό1βˆ’25π‘₯+935π‘₯βˆ’82945π‘₯+57120,790π‘₯βˆ’β‹―οˆ+𝑐π‘₯ο€Ό1βˆ’π‘₯+32π‘₯βˆ’1318π‘₯+119360π‘₯βˆ’β‹―οˆοŠ§οŠ¨οŠ©οŠͺοŠͺ
  • D𝑦=𝑐π‘₯ο€Ό1βˆ’25π‘₯+935π‘₯βˆ’82945π‘₯+57120,790π‘₯βˆ’β‹―οˆ+𝑐π‘₯ο€Ό1βˆ’π‘₯+32π‘₯βˆ’1318π‘₯+119360π‘₯βˆ’β‹―οˆοŠ§οŠ¨οŠ©οŠͺοŠͺ

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