Question

Consider the following initial-value problem. X′=(2 4 −1​ 6​)X,X(0)=(−1 7​) Find the repeated eigenvalue of the coefficient matrix A(t). λ= Find an eigenvector for the corresponding eigenvalue. K= Solve the given initial-value problem. X(t)=___

Answer 1

To find the repeated eigenvalue of the coefficient matrix A(t), we solve the characteristic equation det(A(t) - λI) = 0. The repeated eigenvalue is λ = 4. To find an eigenvector for this eigenvalue, we solve (A(t) - λI)X = 0, which gives us X1 = [2, 1]. Finally, to solve the given initial-value problem X'(t) = (2 4 -1 6)X, X(0) = (-1 7), we use X(t) = e^(At)X(0), where e^(At) is the matrix exponential of A(t).

Step-by-step explanation:

To find the repeated eigenvalue, we need to find the eigenvalues of the coefficient matrix A(t). The eigenvalues are the solutions to the characteristic equation det(A(t) - λI) = 0, where I is the identity matrix.

In this case, the characteristic equation is:

|2-λ 4|

| -1 6-λ| = 0

Simplifying the determinant, we get:

(2-λ)(6-λ) - (4)(-1) = 0

Simplifying further, we get a quadratic equation:

λ^2 - 8λ + 14 = 0

Solving this quadratic equation, we find that the repeated eigenvalue is λ = 4.

To find an eigenvector for this eigenvalue, we need to solve the equation (A(t) - λI)X = 0, where X is the eigenvector.

In this case, the equation is:

|2-4 4|

| -1 6-4| |X1|

Simplifying this equation, we get:

| -2 4||X1| = 0

From this, we can see that the eigenvector is X1 = [2, 1].

Finally, to solve the given initial-value problem X′=(2 4 −1 6)X, X(0)=(−1 7), we can use the formula X(t)=e^(At)X(0), where e^(At) is the matrix exponential of A(t).

Using this formula, we find:

X(t) = e^(4t)|X1|

Substituting in the values we found earlier:

X(t) = e^(4t)|[2, 1]|

Simplifying further:

X(t) = |2e^(4t)|

|1e^(4t)|

Answer 2

To find the repeated eigenvalue of the coefficient matrix A(t), you need to solve the characteristic equation and check if any of the eigenvalues repeat.

Step-by-step explanation:

In order to find the repeated eigenvalue of the coefficient matrix A(t), we need to find the eigenvalues of A(t) and check if any of them repeat. To find the eigenvalues, we need to solve the characteristic equation det(A - λI) = 0, where A is the matrix and λ is the eigenvalue.

Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation A - λI = 0, where I is the identity matrix.

Finally, to solve the initial-value problem X' = A(t)X, X(0) = x, we can use the formula X(t) = e^(At)X(0), where e^(At) is the matrix exponential of A(t).