MATH 20D Lecture 8

Section 4.2 - How to solve

Look for solutions of the form

In general, given the second-order equation whose characteristic equation is . The roots of this characteristic equation is given by

There are now three cases:

1. If then the characteristic equation has two distinct, real-valued roots . The solution to the ODE is
   
   for arbitrary constants .
2. If then the characteristic equation has a repeated root . The solution to the ODE is
   
   for arbitrary constants .
3. If then the characteristic equation has complex roots where and . The solution to the ODE is
   
   for arbitrary constants .

Sections 4.4 & 4.5 - The Method of Undetermined Coefficients (a.k.a. Making educated guesses)

Undetermined Coefficients Method

Consider the non-homogenous second order linear equation with constant coefficients

Then the solution to this non-homogeneous equation, , can be expressed in the form

is soln to the corresponding homogeneous eqn (what we’ve been doing this week)
: any particular solution to the non-hom. eqn

For certain RHS, we can make a guess for what will be.
Here must be one of:

• Exponential:
• Polynomial:
• and/or
• any product/combo of the above 3

$g(t)=k{e}^{\omega t}$

: indep. variable
: arbitrary parameters/constants
Our guess for :

$y^{\prime \prime }-2y^{\prime }-3y=e^{2t},\omega =2$

1. Solve corresponding hom. eqn. for . Here, we solve for
   Chara. eqn.: which has soln
   
   So
2. Guess that , plug in the ODE and solve for .
   and
   plug in in LHS:
   Meanwhile RHS
   So
   Thus,
3. General soln to the non-homo. eqn. is
4. Based on RHS, we guess that . But there is a “conflict” between & so we make another one, by multiplying our current guess by . New guess:
   Now there’s no more “conflict”, we can use and plug it into the ODE to solve for .
   LHS =
   With RHS = , we have so thus,
   General soln: