Worksheet: Modeling with Second-Order Differential Equations

In this worksheet, we will practice using second-order differential equations to model situations in real life.

Q1:

Consider a body of mass ๐‘š attached to a spring with spring constant ๐‘˜ . The differential equation ๐‘š ๐‘ฆ ๐‘ก = โˆ’ ๐‘˜ ๐‘ฆ d d ๏Šจ ๏Šจ , where ๐‘ฆ is the vertical displacement, can be used to model the dynamics of this system. However, such a differential equation implies that, once the body starts moving, it will oscillate forever. A better model is one that also considers the effects of frictional forces. If we add a frictional force that is proportional to the velocity of the body, we get the following differential equation ๐‘š ๐‘ฆ ๐‘ก = โˆ’ ๐‘˜ ๐‘ฆ โˆ’ ๐‘  ๐‘ฆ ๐‘ก , d d d d ๏Šจ ๏Šจ where ๐‘  > 0 .

Letting ๐‘ค = ๐‘˜ ๐‘š ๏Šจ and 2 ๐‘ = ๐‘  ๐‘š , there are three possible behaviours of the solution to this equation which we can define in terms of ๐‘ ๏Šจ and ๐‘ค ๏Šจ as follows: over damped when ๐‘ > ๐‘ค ๏Šจ ๏Šจ ; critically damped when ๐‘ = ๐‘ค ๏Šจ ๏Šจ ; and underdamped or oscillatory when ๐‘ < ๐‘ค ๏Šจ ๏Šจ . Which of the following solutions can be used to describe one of these behaviours:

  1. ๐‘ฆ = ( ๐‘ + ๐‘ ๐‘ก ) ๐‘’ ๏Šง ๏Šจ ๏Šฑ ๏Œป ๏
  2. ๐‘ฆ = ๐‘ ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏Šฑ ๏ฟ ๏ ๏Šจ ๏Šฑ ๏Ž€ ๏ , where ๐œ† = ๐‘ + โˆš ๐‘ โˆ’ ๐‘ค ๏Šจ ๏Šจ and ๐œ‡ = ๐‘ โˆ’ โˆš ๐‘ โˆ’ ๐‘ค ๏Šจ ๏Šจ
  3. ๐‘ฆ = ๐‘ ๐‘’ ( ๐›ฝ ๐‘ก + ๐›พ ) ๏Šฑ ๏Œป ๏ s i n where ๐›ฝ = โˆš ๐‘ค โˆ’ ๐‘ ๏Šจ ๏Šจ
  • Aa
  • Bc
  • Cb
  • DEach solution describes one of the different behaviours

Q2:

The one-dimensional time-independent Schrodinger equation is given as d d ๏Šจ ๏Šจ ๐œ“ ๐‘ฅ = 2 ๐‘š โ„ [ ๐‘ˆ ( ๐‘ฅ ) โˆ’ ๐ธ ] ๐œ“ ,

where ๐œ“ is a wave function which describes the displacement ๐‘ฅ of a single particle of mass ๐‘š , ๐ธ is the total energy, ๐‘ˆ is the potential energy, and โ„ is a known constant. Since ๐‘ˆ ( ๐‘ฅ ) = 0 for the particle-in-a-box model, where 0 โ‰ค ๐‘ฅ โ‰ค ๐‘Ž , this second-order differential equation becomes ๐œ“ โ€ฒ โ€ฒ = ๐›ผ ๐œ“ , ๐›ผ = โˆ’ 2 ๐‘š ๐ธ โ„ . ๏Šจ ๏Šจ ๏Šจ w h e r e

Find the general solution for this differential equation.

  • A ๐‘ฆ = ( ๐‘ + ๐‘ ) ๐‘’ ๏Šง ๏Šจ ๏ต ๏—
  • B ๐‘ฆ = ๐‘ ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏Šจ ๏‘ฎ ๏‘ ๏Žช ๏‘ฎ ๏‘
  • C ๐‘ฆ = ๐‘ ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏ต ๏— ๏Šจ ๏Šฑ ๏ต ๏—
  • D ๐‘ฆ = ( ๐‘ + ๐‘ ) ๐‘’ ๏Šง ๏Šจ ๏Šฑ ๏ต ๏—

Q3:

Consider a mass ๐‘š that oscillates at the end of a spring having a spring constant ๐‘˜ . The following second-order differential equation describes the vertical displacement ๐‘ฆ of this spring-mass system: This differential equation neglects the influence of air resistance or frictional forces. Find the general solution which describes the vertical displacement of this spring as a function of time ๐‘ก .

  • A ๐‘ฆ = ๐‘ ๏€ผ ๐‘ก ๐œ” ๏ˆ + ๐‘ ๏€ผ ๐‘ก ๐œ” ๏ˆ ๏Šง ๏Šจ ๏Šจ ๏Šจ s i n c o s , ๐œ” = ๐‘˜ ๐‘š ๏Šจ
  • B ๐‘ฆ = ๐‘ ( ๐œ” ๐‘ก ) + ๐‘ ( ๐œ” ๐‘ก ) ๏Šง ๏Šจ s i n c o s , ๐œ” = ๐‘˜ ๐‘š ๏Šจ
  • C ๐‘ฆ = ๐‘ ( ๐œ” ๐‘ก ) + ๐‘ ( ๐œ” ๐‘ก ) ๏Šง ๏Šจ ๏Šจ ๏Šจ s i n c o s , ๐œ” = ๐‘˜ ๐‘š ๏Šจ
  • D ๐‘ฆ = ๐‘ ๏€ผ ๐‘ก ๐œ” ๏ˆ + ๐‘ ๏€ผ ๐‘ก ๐œ” ๏ˆ ๏Šง ๏Šจ s i n c o s , ๐œ” = ๐‘˜ ๐‘š ๏Šจ

Q4:

Consider a mass ๐‘š that oscillates at the end of a spring having a spring constant ๐‘˜ . The following second-order differential equation describes the vertical displacement ๐‘ฆ of this spring-mass system: ๐‘š ๐‘ฆ ๐‘ฅ = โˆ’ ๐‘˜ ๐‘ฆ . d d ๏Šจ ๏Šจ

Such a differential equation implies that mass ๐‘š , once started, will simply oscillate up and down forever! This differential equation neglects the influence of frictional forces. Let us assume that there is a retarding force which is proportional to the velocity of motion where ๐‘  is treated as a proportionality constant ( ๐‘  > 0 ) . The aforementioned differential equation now becomes ๐‘š ๐‘ฆ ๐‘ฅ = โˆ’ ๐‘˜ ๐‘ฆ โˆ’ ๐‘  ๐‘ฆ ๐‘ฅ d d d d ๏Šจ ๏Šจ

Let ๐œ” = ๐‘˜ ๐‘š ๏Šจ and 2 ๐‘ = ๐‘  ๐‘š , where ๐‘ > 0 .

There are three possible types of solutions which depend upon the relative size of ๐‘ ๏Šจ and ๐œ” ๏Šจ , including the following:

overdamped if ๐‘ > ๐œ” ๏Šจ ๏Šจ ,

critically damped if ๐‘ = ๐œ” ๏Šจ ๏Šจ ,

underdamped or oscillatory if ๐‘ < ๐œ” ๏Šจ ๏Šจ .

Find the general solution for this differential equation, which describes the vertical displacement of this spring as a function of time ๐‘ก for โ€œoverdampedโ€ motion.

  • A ๐‘ฆ = ๐‘ ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏Šฑ ๏ฟ ๏ ๏Šจ ๏Šฑ ๏Ž€ ๏ , ๐œ† = ๐‘ + โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ , ๐œ‡ = ๐‘ โˆ’ โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ
  • B ๐‘ฆ = ๐‘ ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏Šจ ๏‘‰ ๏‘ธ ๏‘‰ ๏‘น , ๐œ† = ๐‘ + โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ , ๐œ‡ = ๐‘ โˆ’ โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ
  • C ๐‘ฆ = ๐‘ ๐‘ก ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏Šฑ ๏Šจ ๏Šฑ ๏‘‰ ๏‘ธ ๏‘‰ ๏‘น , ๐œ† = ๐‘ + โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ , ๐œ‡ = ๐‘ โˆ’ โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ
  • D ๐‘ฆ = ๐‘ ๐‘ก ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏ฟ ๏ ๏Šจ ๏Ž€ ๏ , ๐œ† = ๐‘ + โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ , ๐œ‡ = ๐‘ โˆ’ โˆš ๐‘ โˆ’ ๐œ” ๏Šจ ๏Šจ

Q5:

Consider a mass ๐‘š that oscillates at the end of a spring having a spring constant ๐‘˜ . The second-order differential equation ๐‘š ๐‘‘ ๐‘ฆ ๐‘‘ ๐‘ฅ = โˆ’ ๐‘˜ ๐‘ฆ ๏Šจ ๏Šจ describes the vertical displacement ๐‘ฆ of this springโ€“mass system.

Such a differential equation implies that mass ๐‘š , once started, will simply oscillate up and down forever. This differential equation neglects the influence of frictional forces. Let us assume that there is a retarding force that is proportional to the velocity of motion where ๐‘  is treated as a proportionality constant ( ๐‘  > 0 ) . The aforementioned differential equation now becomes ๐‘š ๐‘‘ ๐‘ฆ ๐‘‘ ๐‘ก = โˆ’ ๐‘˜ ๐‘ฆ โˆ’ ๐‘  ๐‘ฆ ๐‘ก . ๏Šจ ๏Šจ d d

Let ๐œ” = ๐‘˜ ๐‘š ๏Šจ and 2 ๐‘ = ๐‘  ๐‘š where ๐‘ > 0 .

There are three possible types of solutions that depend on the relative size of ๐‘ ๏Šจ and ๐œ” ๏Šจ , including the following:

overdamped if ๐‘ > ๐œ” ๏Šจ ๏Šจ ,

critically damped if ๐‘ = ๐œ” ๏Šจ ๏Šจ ,

underdamped or oscillatory if ๐‘ < ๐œ” ๏Šจ ๏Šจ .

Find the general solution for this differential equation, which describes the vertical displacement of this spring as a function of time ๐‘ก for the โ€œcritically dampedโ€ motion."

  • A ๐‘ฆ = ( ๐‘ + ๐‘ ) ๐‘’ ๏Šง ๏Šจ ๏Œป ๏
  • B ๐‘ฆ = ๐‘ ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏Šฑ ๏Œป ๏ ๏Šจ ๏Œป ๏
  • C ๐‘ฆ = ( ๐‘ + ๐‘ ๐‘ก ) ๐‘’ ๏Šง ๏Šจ ๏Šฑ ๏Œป ๏
  • D ๐‘ฆ = ๐‘ ๐‘’ + ๐‘ ๐‘’ ๏Šง ๏Šฑ ๏Šจ ๏‘‰ ๏ด ๏‘‰ ๏ด

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