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# Systems of Ordinary Differential Equations

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Solve the matrix differential equation X^'=AX where X= [x_1,〖 x〗_2 ]^T=[■([email protected]_2 )] and A=[■(3&[email protected]&-1)].
Find the eigenvalue(s) of A by solving |λ-A|=0
Solve the linear equation (λ-A)u=0 to get the eigenvector(s) u= 〖[u_1,u_2]〗^2
Find the fundamental matrix Φ(t)
What is the Wronskian for Φ?
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,〖 x〗_2 ]^T=[■([email protected]_2 )] and A=[■(3&[email protected]&-1)].
Find the eigenvalue(s) of A by solving |λ-A|=0
We have

whence the eigenvalues are and .
Solve the linear equation (λ-A)u=0 to get the eigenvector(s) u= 〖[u_1,u_2]〗^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 Φ(t)
The fundamental matrix is a matrix with columns . Thus we have
.
What is the Wronskian for Φ?
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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