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′′=𝑓(𝑦), where 𝑓(𝑦) is defined on the interval π‘Žβ‰€π‘¦β‰€π‘. 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′′=𝑓(𝑦) be rewritten as a system of two first-order differential equations?

  • Add𝑒𝑦=𝑦, dd𝑒π‘₯=𝑓(𝑦)
  • Bdd𝑦π‘₯=𝑒, dd𝑒𝑦=𝑓(𝑦)
  • Cddπ‘₯𝑒=𝑦, dd𝑒π‘₯=𝑓(𝑦)
  • Ddd𝑦π‘₯=𝑒, dd𝑒π‘₯=𝑓(𝑦)

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