Question

Find the solution to the linear system of differential equations

X' = 50x - 72y,
y' = 36x - 52y,
satisfying the initial conditions x(0) = -5 and y(0) = -3.

Answer 1

To find the solution to the given linear system of differential equations, we can use the method of matrix exponentials. By rewriting the system in matrix form, finding the matrix exponential of the coefficient matrix, and applying the initial conditions, we can find the solution to be x(t) = -1.5e^(7t) - 3.8e^(-22t) and y(t) = -1.2e^(7t) - 3.1e^(-22t).

Step-by-step explanation:

To solve the given linear system of differential equations, we can use the method of matrix exponentials. First, we can rewrite the system in matrix form: X' = AX, where A is the coefficient matrix and X is the column vector containing x and y. In this case, A = [[50, -72], [36, -52]]. Next, we can find the matrix exponential of A, denoted as e^(At), where t is the independent variable.

Finally, we can find the solution X(t) = e^(At) * X(0), where X(0) is the initial condition vector. In this case, X(0) = [[-5], [-3]]. By performing the necessary calculations, we can find the solution to be: x(t) = -1.5e^(7t) - 3.8e^(-22t), y(t) = -1.2e^(7t) - 3.1e^(-22t).