Variation of parameters
First, solve the linear homogeneous equation by separating variables. Rearranging terms in the equation gives
Now, the variables are separated, x appears only on the right side, and y only on the left.
Integrate the left side in relation to y, and the right side in relation to x
which is
By taking exponents, we obtain
Hence,we obtain
where
Next, we need to find the particular solution
Therefore, we consider
Let's assume that
since
Therefore
which gives
Now, we can find the function u :
Since we need to find only one function that will male this work, we don’t need to introduce the constant of integration c. Hence,
Recall that
The general solution is
Integrating Factor technique
This equation is linear with
Hence,
So, an integrating factor is
and the general solution is
The integrating factor method, which was an effective method for solving first-order differential equations, is not a viable approach for solving second-order equstions. To see what happens, even for the simplest equation, consider the differential equation
Find solution of