Coefficient of differential

Question

Find a particular solution to each of the following equations trying a solution of the given form and solve for coefficients.

dy/dt + 2y = sin(t). y(t) = Asin(t) + Bcos(t)

For solution i found y = Ae^-2t + 2sin(t)/5 - cos(t)/5. I know this is probably the easier part of the problem, but I am confused on how to solve for the coefficients.

Answer 1

You have already found the solution to the problem. Your solution is A*e^{-2t} + 2*sin(t)/5 - cos(t)/5. This is the most general solution to the differential equation.

The solution to this type of differential equation is broken up into two parts. The "homogeneous solution" and the "particular solution," where the overall solution is the sum of the particular and homogeneous solutions.

The homogeneous solution is the solution to the equation dy/dt +2y = 0, that is, the solution you get if you set anything that is not multiplying a "y" or "dy/dt" to 0. The homogeneous solution to this equation is A*e^{-2t}.

The particular solution is whatever part of the solution is NOT covered by the homogeneous solution. To find the particular solution to the equation, you just ignore the A*e^{-2t} term.

The particular solution is y = 2*sin(t)/5 - cos(t)/5. It is called the "particular solution" because it does not contain any arbitrary constants.

The problem tells you that the form of the particular solution is A*sin(t) + B*sin(t), and in solving the differential equation, you found that A = 2/5 and B=-1/5.

In summary, you basically already solved the problem, but didn't know it yet.

Let me know if you have any questions about the terminology. Also, how did you solve this equation? Did you use the integrating factor method, or undetermined coefficients?