Solve the linear equations by considering y as a function of x, that is,
Solve the linear equations by considering y as a function of x, that is,
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
Let’s solve the integral on the right side.
Therefore
By taking exponents, we obtain
Hence,we obtain
where
Next, we need to find the particular solution
Therefore, we consider
Let’s assume that
Substituting
Therefore,
which gives
Now, we can find the function u:
Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
where we assume s is a positive real number. For example, to find the Laplace transform of
Verify the following Laplace transforms, where u is a real number.
Find Laplace transform of the given function