Final answer:
To solve the non-homogeneous second-order differential equation, we find the complementary solution (homogeneous solution) and particular solution, then combine them to get the general solution. Initial conditions, if given, would be required to solve for constants in the complementary solution.
Step-by-step explanation:
The student is asking for the general solution of a non-homogeneous second-order differential equation with variable coefficients. To solve this differential equation, we first find the complementary solution (homogeneous solution) by setting the right-hand side to zero, so that y'' + y = 0. The general solution to this is in the form of a linear combination of sines and cosines, which is Yh(t) = A cos(t) + B sin(t), where A and B are constants.
To find the particular solution, we use the method of undetermined coefficients. The right-hand side of the equation, 7sin²(t) + tcos²(t), suggests a particular solution of the form Yp(t) = At sin(t) + Bt cos(t) combined with terms that are second-order polynomials multiplied by sine or cosine to accommodate the t in the non-homogeneous term. Once the particular solution is found, the general solution of the differential equation is the sum of the homogeneous and particular solutions: .
We must also apply the initial conditions if they are provided to solve for constants A and B in the complementary solution, but these are not specified in the question. In summary, determining the constants would involve plugging the initial conditions into the general solution and solving for the constants.