Exercise 2.6
1. Solve the following Euler's equation. provided .
(a)
(b)
(c)
(d)
(e)
Answer
1.
(a) Set
. Then the indicial equation is
Thus, we have
. Note that this indicial equation is the characteristic equation of the following differential equation.
Therefore, the general solution is
(b) Set
. Then the indicial equation is given by
Thus, we have
. Note that this indicial equation is the characteristic equation of the following equation.
Therefore, the general solution is given by
(c) Set
. Then the indicial equation is given by
Thus, we have
. Note that this indicial equation is the characteristic equation of the following equation.
Thus the complementary function is
We find the particular solution. Since
, we have
which implies the solutions are
. But
are already used in the . So, we omit these solutions from . Then we have
Substitute this into *. Then
Thus we have
and ,
Therefore, the general solution is
(d) Set
. Then the indicial equation is given by
Simplifying to get
Thus,
. This indicial equation is the characteristic equation of the following differential equation.
Thus the complementary function is given by
Now we find the particular solution using the variation of parameter. Since
, we have
Solve this for to get
. But
are already used in . So, we omit from . Then we have
Substitute this into *.
Then
and
Therefore, the general solution is given by
(e) Given equation is not Euler's equation. But if we multiply to both sides. Then it becomes Euler's equation.
Now let
. Then the indicial equation is
Simplify this equation. We get
Thus we have
. Not that this indicial equation is th echaracteristic equation of the following differential equation.
Therefore, the general solution is