Solve the following differential equation?
(D-1)^(2)[D^(2)+1)^(2)y=sin^(2)(x/2)+e^(x)+x
(D-1)^(2)[D^(2)+1)^(2)y=sin^(2)(x/2)+e^(x)+x
1 Answer
for constants
Explanation:
We can rewrite this as
Just looking at the left side,
(where we define
Let's focus on the specific solution before we discuss homogenous solutions. It is logical to assume some functional forms of our solutions.
For the constant, clearly that constant can be our function, i.e.
For a linear function, all higher order derivatives will be negligible, so we can try a form of
To get the constants above, therefore
For an exponential function
Therefore, we try
Therefore (the
Therefore, we will try
Therefore, the first two terms will cancel out again, and
Therefore,
We can repeat this process for the sinusoid and we find that we need to use
As is apparent by the factored differential equation, we have eigenfunctions of
Therefore, our general solution is