Worksheet: Reduction of Order for Second-Order Differential Equations

In this worksheet, we will practice using reduction of order to find a second solution to a second-order differential equation given the first solution.

Q1:

Consider 𝑦 β€² β€² = 𝑓 ( 𝑦 ) , where 𝑓 ( 𝑦 ) is defined on the interval π‘Ž ≀ 𝑦 ≀ 𝑏 . It is possible to rewrite this second-order differential equation as a system of two first-order differential equations using an appropriate change of variable. Using 𝑒 as the new variable, how can 𝑦 β€² β€² = 𝑓 ( 𝑦 ) be rewritten as a system of two first-order differential equations?

  • A d d 𝑒 𝑦 = 𝑦 , d d 𝑒 π‘₯ = 𝑓 ( 𝑦 )
  • B d d 𝑦 π‘₯ = 𝑒 , d d 𝑒 π‘₯ = 𝑓 ( 𝑦 )
  • C d d π‘₯ 𝑒 = 𝑦 , d d 𝑒 π‘₯ = 𝑓 ( 𝑦 )
  • D d d 𝑦 π‘₯ = 𝑒 , d d 𝑒 𝑦 = 𝑓 ( 𝑦 )

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