Final answer:
To find a particular solution to the differential equation y'' + 25y = 10sec(5t), we use the method of undetermined coefficients and assume a particular solution of the form Y_p = Asec(5t) + Bcos(5t) + Csin(5t). Plugging this into the differential equation, we find the values of the constants A, B, and C and substitute them back into the particular solution to obtain the final answer.
Step-by-step explanation:
To find a particular solution to the differential equation y'' + 25y = 10sec(5t), we can use the method of undetermined coefficients. Since the right-hand side of the equation is 10 times the secant function, we can assume that the particular solution has the form Y_p = Asec(5t) + Bcos(5t) + Csin(5t), where A, B, and C are constants.
Plugging this into the differential equation, we can find the values of A, B, and C. Differentiating twice and substituting back into the equation will give three equations which can be solved to find the values of the constants.
After solving for the constants, we can substitute them back into the particular solution to get the final answer: Y_p = 10/25sec(5t) - 2/5cos(5t) + 0sin(5t).