Special Second Order Differential Equations

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Special Second Order Differential Equations

• Eric, Robert, Nathan

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Special Second Order Differential Equations

• Two categories
• 1) Equations with the Dependent Variable Missing
  yf(t,y)
• 2) Equations with the Independent Variable
  Missing yf(y,y)

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1) Equations with the Dependent Variable Missing

• Of the form yf(t,y)
• One can make substitutions to yield a first order
  differential equation
• (1)Substitutions being
• vy
• vy
• (2)Post substituting yields a DE of the form
  vf(t,v)

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Equations with the Dependent Variable Missing
(cont)

• (3)If the prior equation can be solved for v then
  y may be obtained by integrating dy/dtv
• (4)Note the 2 constants of integration obtained
  when solving the first order DE, c1, then another
  in the integrations of y, c2.
• This will become more clear with Roberts example.

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2) Equations with the Independent Variable Missing

• Of the form yf(y,y)
• Let vy
• Thus vy
• dv/dt(dv/dy)(dy/dt)
• (dy/dt)v
• Thus dv/dtv(dv/dy)
• dv/dtvy
• (1) Substitutions are
• yv(dv/dy)
• yv

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Equations with the Independent Variable Missing
(cont)

• (2) Substitutions give a DE of the form
  v(dv/dy)f(y,v), which is 1st order.
• (3) If the above 1st order can be solved, this
  gives vf(y), recall vy
• Then you have a Separable DE
• (4) Note the 2 constants of integration obtained
  when solving the first order DE, c1, then another
  from the separable DE, c2.
• This will also become more clear with Erics
  Example

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Homework

• Page 133
• Numbers (40,42)