Find the general solution of y"-3y=8e^(3t)+4sint

To find the general solution of the differential equation y'' - 3y = 8e^(3t) + 4sin(t), we can follow these steps:

Step 1: Find the complementary function.
- To find the complementary function, we consider the homogeneous part of the equation y'' - 3y = 0.
- The characteristic equation is obtained by substituting y = e^(rt) into the homogeneous equation: r^2 - 3 = 0.
- The roots of the characteristic equation are r1 = √3 and r2 = -√3.
- This gives us the complementary function: yc = c1e^(√3t) + c2e^(-√3t), where c1 and c2 are constants.

Step 2: Find a particular solution.
- To find a particular solution yp, we need to consider the particular non-homogeneous terms.
- For the term 8e^(3t), we try a particular solution of the form yp = Ae^(3t), where A is a constant to be determined.
- Differentiating yp twice and substituting it into the original equation, we get 9Ae^(3t) - 3Ae^(3t) = 8e^(3t).
- Solving for A, we find A = 1.
- Therefore, a particular solution for the term 8e^(3t) is yp = e^(3t).

- For the term 4sin(t), we try a particular solution of the form yp = Bsin(t) + Ccos(t), where B and C are constants to be determined.
- Differentiating yp twice and substituting it into the original equation, we get -Bsin(t) - Ccos(t) - 3(Bsin(t) + Ccos(t)) = 4sin(t).
- Equating the coefficients of sin(t) and cos(t) separately, we get -4B - 3C = 0 and -4C + 3B = 4.
- Solving these linear equations, we find B = -2/5 and C = -3/5.
- Therefore, a particular solution for the term 4sin(t) is yp = (-2/5)sin(t) - (3/5)cos(t).

Step 3: Find the general solution.
- The general solution of the differential equation is given by y = yc + yp.
- Combining the complementary function and the particular solutions, we have:
y = c1e^(√3t) + c2e^(-√3t) + e^(3t) - (2/5)sin(t) - (3/5)cos(t).

And that's the general solution of the given differential equation.