# Systems of Ordinary Differential Equations

Solve the matrix differential equation X^'=AX where X= [x_1,〖 x〗_2 ]^T=[■(x_1@x_2 )] and A=[■(3&-1@-5&-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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#### Solution Preview

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Solve the matrix differential equation X^'=AX where X= [x_1,〖 x〗_2 ]^T=[■(x_1@x_2 )] and A=[■(3&-1@-5&-1)].

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

We have

whence the ...

#### Solution Summary

We solve systems of ordinary differential equations by using matrices.

$2.19