c) Find a particular solution satisfying the initial condition
Compute the RHS and compare it with the LHS
Define the RHS
Compute the LHS
Define the LHS
Define the components of the derivative of the solution
b) Verify that the general solution satisfies the system
Display the general solution
Define the general solution
Define and compute the second solution
Define and compute the second solution
Define and compute the first solution
We have to redefine the general solution using different constants.
Another way
Thus
Compute the rref of the aug. matrix
Create the augmented matrix and name it Xb
Define the vector with initial conditions
where X(0) is the matrix with columns x1(0), x2(0) and x3(0). Thus, to solve the system we form the augmented matrix [ x1(0) x2(0) x3(0) | b ] , where b = and compute its rref
To find the constants c1, c2 and c3, we need to solve the nonhomogeneous system
Compute the rref of
To find the e-vectors, we need to solve for v
Or, by using the eigenvals command
Compute the eigenvalues by solving the characteristic equation
Define the 3 x 3 identity
Define A
Set the origin to 1
Solution:
a) Solve the system
Remark
This worksheet can be used to solve any system with an eigenvalue having multiplicity 3 and defect 2.
Example (Problem 25) Multiplicity 3, defect 2
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Use v2 to computer v1 provided that v1 is not the zero vector
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Use v3 to computer v2 provided that v2 is not the zero vector
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Let
all three components of v3 are arbitrary
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Compute the rref
So, we find v3 by solving the system
From the rref we notice that the e-value is able to produce only one e-vector. That is, the e-values is defective with defect equals 2