Obtain the general solution of the differential equations by using the method of undetermined coefficients
y''' - 2y'' + 3y' - 5y = 5sin2x + 10x^2 - 3x - 7
y''' + y' - 5y = 4sinx + 2x^2
Those are not easy to find but it is clear that neither 0 nor 2i is a root so you need to look for a "particular solution" of the form
y= Acos(2x)+ Bsin(2x)+ Cx^2+ Dx+ E
Here the characteristic equation is . Again, that has only irrational roots which will be hard to find but we know the "particular solution" must be of the formy''' + y' - 5y = 4sinx + 2x^2
Are you sure you have copied the problems correctly? It looks to me like the general solution to either of those will be very hard to find.
Let's look at the first one:
The differential operator annihilates and annihilates hence the operator:
Thus, applying to both sides of (1) gives us:
The characteristic roots are then:
where is of multiplicity 4 and are of multiplicity 2, and so the general solution to (2) is given by:
Now, recall that a general solution to (1) is of the form . Since every solution to (1) is also a solution to (2), then must have the form displayed on the right-hand side of (3). But we recognize that:
and so there must exist a particular solution of the form:
Now it is a matter of using the method of undetermined coefficients to determine the particular solution that satisfies (1).
So, what you need to do is compute and , substitute these into (1), and solve the resulting linear system that arises when you equate coefficients. Can you proceed?