# Find #f# ?

##
#f''(x)+f(x)=1# ,

#x# #in# #RR#

##### 2 Answers

#### Explanation:

The quickest way of solving this equation is to note that the function

satisfies

or

This is a familiar equation with the well known solution

where

#### Explanation:

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

Denoting

This is easily solved by introducing the integrating factor

where

But

This equation has an integrating factor

This is easily integrated to

leading 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