Find #f# ?
#x# #in# #RR#
The quickest way of solving this equation is to note that the function
This is a familiar equation with the well known solution
If you want a quick solution, see https://socratic.org/s/aQcKQPv8
I will describe a more detailed solution here. This involves much more work, but has the advantage of being a much more general method. It is also a simple introduction to the factorization method for solving differential equations.
Let us denote the operator
We can write
where the use of the familiar formula
This is easily solved by introducing the integrating factor
This equation has an integrating factor
This is easily integrated to
This is the required solution, but we have one more step to go. The solution has to be real, and so must be equal to its complex conjugate. It is easy to see that this means that
the solution becomes