The student is seeking help with a differential equation, specifically finding the particular solution to y''+25y = -30sin(5t). The method of undetermined coefficients is applied by guessing a solution of Acos(5t) + Bsin(5t) and determining A and B by equating coefficients.
The student is asking for help in finding a particular solution to a second-order differential equation 'y''+25y = -30sin(5t)'. To find the particular solution (yp), we can assume a solution of the form Acos(5t) + Bsin(5t) since the non-homogeneous part is a sine function. We substitute our assumed solution into the differential equation and solve for A and B. This technique is known as the method of undetermined coefficients and is used when the non-homogeneous term is a simple trigonometric, exponential, or polynomial function.
After plugging the assumed solution and its derivatives into the original equation, we equate coefficients of cos(5t) and sin(5t) on both sides to find values for A and B. These coefficients A and B will give us the particular solution to the differential equation that satisfies the non-homogeneous part.
The answer will be in the form yp = Acos(5t) + Bsin(5t) after determining the values of A and B.