Final answer:
To solve the initial value problem x' = [3 -10; 1 -3]x, x(0) = [4; 3], use the method of matrix exponential. The solution is x(t) = [11/4 * e^(3t) + 13/4 * e^(-3t); 3/4 * e^(3t) + 5/4 * e^(-3t)].
Step-by-step explanation:
To solve the initial value problem x' = [3 -10; 1 -3]x, x(0) = [4; 3], we can use the method of matrix exponential. The solution is given by x(t) = e^At * x(0), where e^At is the matrix exponential and A is the given matrix.
To calculate the matrix exponential, we first need to find the eigenvalues and eigenvectors of A. Then we can form the diagonal matrix D = [lambda1 0; 0 lambda2], where lambda1 and lambda2 are the eigenvalues. Finally, we can calculate e^At = P * e^Dt * P^(-1), where P is the matrix of eigenvectors.
Substituting the values into the formula, we get x(t) = [11/4 * e^(3t) + 13/4 * e^(-3t); 3/4 * e^(3t) + 5/4 * e^(-3t)].