The student's question involves solving a boundary value problem using a finite difference method for the approximation of the second derivative. By discretizing the interval and using the provided boundary conditions, a system of linear equations is created and solved to find the values of the function y(t).
To solve the boundary value problem (BVP) using the given approximation for y''(t), we first substitute into the differential equation y'' = 4y to get the difference equation:
y(t+h) - 2y(t) + y(t-h) = 4h^2y(t)
This is a second order linear difference equation which needs two boundary conditions to solve. Given y(0) = 1 and y(1) = 3, we can use these to find the values of y at points t = h, 2h, ..., 1-h. We divide the interval [0,1] into n equal parts, so that h = 1/n. By applying the boundary conditions and solving the system of equations obtained from the difference equation, we find the values of y at the discrete points, then we can interpolate to find y(t) for any t in the interval.
In conclusion, the final answer is obtained by finding the solution to the system of linear equations that arise from the discretization of the differential equation with the provided boundary conditions.