In this section we want to look for solutions to


(1)

around . These type of differential equations are
called Euler Equations.
Recall from the previous section
that a point is an ordinary point if the quotients,
have Taylor
series around . However, because of the x in the denominator neither of these will have a Taylor series around and so is a singular point. So, the method from the previous section
won’t work since it required an ordinary point.
However, it is possible to get solutions to this
differential equation that aren’t series solutions. Let’s start off by assuming that x>0 (the reason for this will be
apparent after we work the first example) and that all solutions are of the
form,
Now plug this into the differential equation to get,
Now, we assumed that x>0
and so this will only be zero if,
So solutions will be of the form (2)
provided r is a solution to (3). This equation is a quadratic in r and so we will have three cases to
look at : Real, Distinct Roots, Double Roots, and Complex Roots.
Real, Distinct Roots
There really isn’t a whole lot to do in this case. We’ll get two solutions that will form a fundamental set of solutions (we’ll
leave it to you to check this) and so our general solution will be,
Example 1 Solve
the following IVP
Solution
We first need to find the roots to (3).
The general solution is then,
To find the constants we differentiate and plug in the
initial conditions as we did back in the second order differential equations
chapter.
The actual solution is then,

With the solution to this example we can now see why we
required x>0. The second term would have division by zero
if we allowed x=0 and the first term
would give us square roots of negative numbers if we allowed x<0.
Double Roots
This case will lead to the same problem that we’ve had every
other time we’ve run into double roots (or double eigenvalues). We only get a single solution and will need a
second solution. In this case it can be
shown that the second solution will be,
and so the general solution in this case is,
We can again see a reason for requiring x>0. If we didn’t we’d
have all sorts of problems with that logarithm.
Example 2 Find
the general solution to the following differential equation.
Solution
First the roots of (3).
So the general solution is then,

Complex Roots
In this case we’ll be assuming that our roots are of the form,
If we take the first root we’ll get the following solution.
This is a problem since we don’t want complex solutions, we
only want real solutions. We can
eliminate this by recalling that,
Plugging the root into this gives,
Note that we had to use Euler formula as well to get to the
final step. Now, as we’ve done every
other time we’ve seen solutions like this we can take the real part and the
imaginary part and use those for our two solutions.
So, in the case of complex roots the general solution will
be,
Once again we can see why we needed to require x>0.
Example 3 Find
the solution to the following differential equation.
Solution
Get the roots to (3)
first as always.
The general solution is then,

We should now talk about how to deal with x<0 since that is a possibility on
occasion. To deal with this we need to
use the variable transformation,
In this case since x<0
we will get η>0.
Now, define,
Then using the chain rule we can see that,
With this transformation the differential equation becomes,
In other words, since η>0 we can use the work above to get
solutions to this differential equation.
We’ll also go back to x’s by
using the variable transformation in reverse.
Let’s just take the real, distinct case first to see what
happens.
Now, we could do this for the rest of the cases if we wanted
to, but before doing that let’s notice that if we recall the definition of
absolute value,
we can combine both of our solutions to this case into one
and write the solution as,
Note that we still need to avoid x=0 since we could still get division by zero. However this is now a solution for any
interval that doesn’t contain x=0.
We can do likewise for the other two cases and the following
solutions for any interval not containing x=0,.
We can make one more generalization before working one more
example. A more general form of an Euler
Equation is,
and we can ask for solutions in any interval not containing . The work for generating the solutions in this
case is identical to all the above work and so isn’t shown here.
The solutions in this general case for any interval not
containing x=a are,
Where the roots are solutions to
Example 4 Find
the solution to the following differential equation on any interval not
containing x=6.
Solution
So we get the roots from the identical quadratic in this
case.
The general solution is then,
