Ex 2. Linear ODE

Calculus 2



Weiqi Zhou

First Order Linear ODE

Homogeneous Linear ODE

  • The ODE is called homogeneous if .

  • It is homogeneous in , i.e., the equation is invariant under

  • The map is linear

Structure of Solutions

  • Given and

  • If is a solution of the homogeneous equation，then so is

  • If is a solution of the non-homogeneous equation，then so is

  • If are solutions of the non-homogeneous equation，then is a solution of the homogeneous equation

Approach

  • Separate variables for the homogeneous equation :

  • Let



  • plug back in to get , and integrate to get

Exmaple

  • Find the solution to

  • Solutions to :



  • Solution：

Exercises

  • Find solutions to

  • Find solutions to
    (hint：)

  • Find solutions to
    (hint：take )

  • Find the solution to with

Second Order Linear ODE

High Order Linear ODE

Linear Independence

  • Given ，if there exist , at least one of which is non-zero, and
    holds for all ，then are linearly dependent on

  • If the equality holds only if ，then are linearly independent on

Linear Combinations

Given ，and ，the expression

is called a linear combination of (w.r.t. )

Linear Representations

  • Example：， are linearly dependent on

  • Example：are linearly independent on

  • Linear dependence implies that one member (with non-zero coefficients) can be represented by linear combinations of other members
    i.e., if , then

Structure of Solutions

  • Given a homogenous linear ODE

  • If are its linearly independent solutions，then its solution set consists of all possible linear combinations of

  • i.e., any of its solutions can be written as for some

Structure of Solutions

  • Given a non-homogenous linear ODE

  • If is a solution of it and is the solution set of

  • then its solution set is

Exercises

  • Verify that are solutions to ，and find the solution set of this equation

  • Suppose that are linearly independent solutions to ，represent other solutions of the euqation by linear combinations of

  • If are solutions to ，find the paricular solution of this equation that satisfies

Summary

  • Homogeneity and Linearity

  • Structure of Solutions

  • Solving First Order Linear ODE