Systems of Ordinary Differential Equations
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
https://brainmass.com/math/ordinary-differential-equations/systems-ordinary-differential-equations-444172
SOLUTION This solution is FREE courtesy of BrainMass!
Please see the attachment.
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.
© BrainMass Inc. brainmass.com December 24, 2021, 10:06 pm ad1c9bdddf>https://brainmass.com/math/ordinary-differential-equations/systems-ordinary-differential-equations-444172