Final answer:
To find the general solution of the given differential equation, solve the homogeneous equation and find a particular solution, then sum them. This requires the use of the characteristic equation and possibly methods like undetermined coefficients or variation of parameters.
Step-by-step explanation:
The given differential equation is 1/4(y)'' + 4y = Tan4t - 31e²ᵗ. To find the general solution, you would solve the corresponding homogeneous equation 1/4(y)'' + 4y = 0 and find a particular solution to the non-homogeneous equation. The homogeneous solution can be found using methods such as the characteristic equation or using the ansatz y(t) = e^{λt}. For the particular solution, you can attempt methods like undetermined coefficients or variation of parameters, taking into account the non-homogeneous terms Tan4t and 31e²ᵗ.
Once you have both the homogeneous solution y_h(t) and a particular solution y_p(t), the general solution is y(t) = y_h(t) + y_p(t). It is left as an exercise to the student to find the explicit forms of these solutions and prove that their sum is, in fact, the correct general solution to the differential equation by plugging them back into the original equation and verifying it satisfies the equation.