Find y particular of the ODE
y"+y'=x+sin(2x)
Apply your methods and see whether you can arrive at the answer from Wolframalpha.com:
http://www.wolframalpha.com/input/?i=y%22%2By%27%3Dx%2Bsin%282x%29
To find the particular solution of the ordinary differential equation (ODE) y'' + y' = x + sin(2x), we can use the method of undetermined coefficients. This method involves guessing the form of the particular solution based on the non-homogeneous term of the equation.
1. Find the complementary solution:
First, we find the complementary solution by solving the homogeneous equation, which is obtained by setting the non-homogeneous term (x + sin(2x)) to zero:
y'' + y' = 0
The characteristic equation is obtained by assuming y = e^(rx):
r^2 + r = 0
Factoring out r, we get:
r(r + 1) = 0
So the solutions for the complementary solution are r = 0 and r = -1. Therefore, the complementary solution is of the form:
y_c = c1e^(-x) + c2
2. Guess the form of the particular solution:
The particular solution, denoted as y_p, should have a similar form to the non-homogeneous term. In this case, x + sin(2x) is a combination of a linear term and a sine term.
To match the linear term, we guess that the particular solution has the form A1x + A0, where A1 and A0 are constants.
To match the sine term, we guess that the particular solution has the form Bsin(2x) + Ccos(2x), where B and C are constants.
3. Find the particular solution:
Substitute the guessed particular solution into the ODE and simplify:
y_p'' + y_p' = (A1 - 4Bcos(2x) - 4Csin(2x)) + (A1 - 2Bsin(2x) + 2Ccos(2x))
The right-hand side of the ODE is x + sin(2x), so we set up the following system of equations to find the constants A1, A0, B, and C:
A1 - 4B = 1
-2B + 2C = 1
Solving this system of equations will give the values of A1, A0, B, and C, which are the coefficients of the particular solution.
4. Combine the complementary and particular solutions:
Once you have the constants A1, A0, B, and C, you can write the particular solution as:
y_p = A1x + A0 + Bsin(2x) + Ccos(2x)
The general solution of the ODE is the sum of the complementary solution and the particular solution:
y = y_c + y_p = c1e^(-x) + c2 + A1x + A0 + Bsin(2x) + Ccos(2x)
This is the particular solution of the given ODE.