Final answer:
To find a particular solution for the differential equation y'' + 25y = 40sin(5t), we can use the method of undetermined coefficients. The particular solution is y = (8/5)cos(5t).
Step-by-step explanation:
To find a particular solution to the differential equation y'' + 25y = 40sin(5t), we can use the method of undetermined coefficients. Since the right side of the equation is 40sin(5t), we can guess that the particular solution has the form y = Asin(5t) + Bcos(5t). Substituting this into the equation, we get -25Asin(5t) + 25Bcos(5t) + 25Asin(5t) + 25Bcos(5t) = 40sin(5t).
Combining the terms with sin(5t), we obtain 50Bcos(5t) = 40sin(5t).
Therefore, B = 8/5 and A = 0. The particular solution is y = (8/5)cos(5t).