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