Final answer:
The general solution to the differential equation y'' + 25y = 11sec²(5t) involves finding the homogeneous solution and a particular solution using an ansatz related to the non-homogeneous term, and combining them.
Step-by-step explanation:
To solve the differential equation y'' + 25y = 11sec²(5t), we first look for the homogeneous solution, which is the general solution to the associated homogeneous equation y'' + 25y = 0. The characteristic equation for this is λ² + 25 = 0, which has solutions λ = ±5i. Therefore, the general homogeneous solution is ψh(t) = C1*cos(5t) + C2*sin(5t), where C1 and C2 are constants.
To find a particular solution ψp(t) to the non-homogeneous equation, we use an ansatz (educated guess) that since the non-homogeneous term is 11sec²(5t), a multiple of a trigonometric function, we can try a solution of the form Atan(5t) + B, where A and B are constants to be determined.
After finding the particular solution by substituting the ansatz into the differential equation and solving for A and B, we combine the homogeneous and particular solutions to form the general solution, ψ(t) = ψh(t) + ψp(t). This process involves differentiating the ansatz to match the given non-homogeneous term and solving for A and B by equating coefficients. The final form of the solution will depend on the specific values of A and B found.