Final answer:
To find a particular solution to the given second-order linear homogeneous differential equation, we can use the method of undetermined coefficients by assuming the particular solution has the same form as the non-homogeneous term and determine the coefficients by substitution.
Step-by-step explanation:
To find a particular solution to the given differential equation, we can use the method of undetermined coefficients. In this method, we assume that the particular solution has the same form as the non-homogeneous term of the equation and determine the coefficients by substitution.
In this case, the non-homogeneous term is (15.5e^t(3t))/(t^(2)+1), which can be written as 15.5e^t(3t)/(t^2+1) or 15.5e^t(3t)(t^2+1)^(-1).
Now, we assume the particular solution is of the form y_p(t) = Ate^t, where A is a constant to be determined.
By substituting this assumed solution into the differential equation and solving for the coefficient A, we can find the particular solution.