# Systems of Ordinary Differential Equations

Solve the matrix differential equation X^'=AX where X= [x_1,ã€– xã€—_2 ]^T=[â– ([email protected]_2 )] and A=[â– (3&[email protected]&-1)].

Find the eigenvalue(s) of A by solving |Î»-A|=0

Solve the linear equation (Î»-A)u=0 to get the eigenvector(s) u= ã€–[u_1,u_2]ã€—^2

Find the fundamental matrix Î¦(t)

What is the Wronskian for Î¦?

Use the result from a to c to express the general solution

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Solve the matrix differential equation X^'=AX where X= [x_1,ã€– xã€—_2 ]^T=[â– ([email protected]_2 )] and A=[â– (3&[email protected]&-1)].

Find the eigenvalue(s) of A by solving |Î»-A|=0

We have

whence the eigenvalues are and .

Solve the linear equation (Î»-A)u=0 to get the eigenvector(s) u= ã€–[u_1,u_2]ã€—^2

The eigenvector may be solved from the equation

Thus we have , whence

is an eigenvector of A with corresponding eigenvalue 4.

The eigenvector may be solved from the equation

Thus we have , whence

is an eigenvector of A with corresponding eigenvalue .

Find the fundamental matrix Î¦(t)

The fundamental matrix is a matrix with columns . Thus we have

.

What is the Wronskian for Î¦?

The Wronskian is given by

Use the result from a to c to express the general solution

The general solution to the system is given by

where and are arbitrary constants.

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