MATH 20D Lecture 12

A second-order linear ODE with variable-coefficients is given by

where is non-zero. This can also be converted into standard form

Bad news: There is no method to solving the general case above!
What we can do:

1. If we have 2 linearly independent solutions to the corresponding homogeneous case
   
   (call them &) then .
   Then use variation by parameters to find
2. If we have only one solution to the given ODE (call )
   Then we can use Reduction by Order to generate that is independent from
   Now go back to case 1
3. When the coefficients are “nice” then we can still find .
   An example of “nice coefficient” is Cauchy-Euler Equation
   where , and are constants.

Cauchy-Euler Equation

where are constants.
The solutions to the corresponding homogeneous equation can be found in the form for .
Here then ,
So LHS=



when RHS: characteristic equation

$t^2{y}^{\prime \prime }+14t{y}^{\prime }+42y=0$

which has 2 soln so
and are 2 linearly independent solutions
Thus

Cases of the Cauchy-Euler Equation

1. If there are two distinct, real-valued solutions . Then the general solution is given by
2. If there are complex solutions . Then the general solution is given by
3. If there is a repeated/unique solution then the general solution is given by
   Remark: Homogeneous Cauchy-Euler equation can be converted into a 2nd order linear ODE with constant coefficients under the substitution