Boundary Value Problems/Review: ODEs

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For more of an introduction to ODEs you are referred to Boundary Value Problems/BVP-Ordinary-Differential-Equations and for examples please look at Examples of ordinary differential equations

First order differential equation:

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Let   the equation

 

is a linear ordinary (single independent variable) differential equation of first order (first derivative). If   on   the equation is called "homogeneous" differential equation. Otherwise it is called nonhomogeneous otherwise.

Examples of homogeneous and non-homogeneous:

  where   is a constant coefficient.

  where   is a coefficient that increases as   increases.

  where  .

Solving homogeneous ODE with constant coefficients: Separable case

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Solving  

 

 

 

where   is the integration constant.

 

 

 

After integration there is a constant, an unknown. This will happen each time you integrate. To determine this unknown for a particular application you will need another piece of information. Typically it is another equation that requires  . This condition with the above differential equation defines an Initial Value Problem. For this problem the value of  is determined by using an intial condition such as  . Then   and making the substitution provides the solution  

Student work

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Solve the following IVPs:

  1. Differential eq:   where   for all  . Initial value:  

Click here for solution: Boundary Value Problems/BVP-solution-1


  1. Differential eq:   where   for all  . Initial value:  

Click here for solution: Boundary Value Problems/BVP-solution-2



Pre-class work:

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Complete each of the following:

Second order Differential Equations

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Cauchy Euler

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Bessel Equation

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