What is the form that I'm supposed to turn this g(t) into for non homogeneous equations

Question

g(t) = 8sin(8t) 

I was thinking the form Asin(8t) + Bcos(8t)

 

But the solution shows a form in A*t*cos(8t)

Answer 1

It depends on the form of the inhomogeneous term of the equation.  The general solution is the sum of the solution to the associated homogeneous equation and a particular solution to the inhomogeneous part.

 

The form

 

g(t) = Asin(8t) + Bcos(8t)

 

is a general solution to the homogeneous equation 

 

g"(t) + 64g(t)  = 0

 

But if you started with the homogeneous equation

 

x"(t) + 64x(t) = f(t)

 

then you would have to add what is called a particular solution x_p(t) for the inhomogeneous term f(t) ≠ 0.

 

The general solution then becomes

 

x(t) = Asin(8t) + Bcos(8t) + x_p(t)

 

where x_p(t) depends on the form of f(t).  Textbooks on differential equations usually have tables of x_p(t) forms for different forms of f(t).

 

This is a lot easier to explain by working an example.  I suggest you post a complete example of a second order inhomogeneous differential equation with constant coefficients, then a tutor can work the solution and this will become more clear.  It is hard to just give a general answer.  It changes with every form of inhomogeneous term f(t).