solve the differential equation y'' - 3y' - 4y = 2 sin x using method of undetermined coefficients.
Solution:
Let y = A sin x + B cos x be the assumed solution.
Substituting this into the differential equation, we get:
A sin x + B cos x - 3(A cos x - B sin x) - 4(A sin x + B cos x) = 2 sin x
Simplifying, we get:
A - 3B - 4A = 2
B + 3A - 4B = 0
Solving the system of equations, we get:
A = 2
B = -1
Therefore, the solution to the differential equation is:
y = 2 sin x - cos x