<Text-field style="Heading 1" layout="Heading 1">Example of Solving Nonhomogeneous Linear System with Complex Eigenvalues</Text-field>

Exercise 14, page 556, Section 9.7

<Text-field style="Heading 2" layout="Heading 2">Maple's Solution to Check Against</Text-field>

Set up the two equations with initial conditions and solve with dsolve

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PkkkZXEyRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YkNiNJInRHRiRGLSwmLUkieEdGJEYsIiIiRjFGMQ==

PkkoZGlmZmVxc0c2IjYkSSRlcTFHRiRJJGVxMkdGJA==

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print(); # input placeholder

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Step by Step Solution using Method of Eigenvalues and Variation of parameters:</Text-field>

First find the general solution to the related homogeneous system

Step 1: Form the matrix of coefficients A for the system

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Step 2: Find the eigenvalues and corresponding eigenvectors for A

First column of vc is eigenvector corresponding to eigenvalue in first entry in vl, etc.

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Step 3: Form the general homogeneous solution ( xh ) using these:

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PkkjeGhHNiIsJiomSSNjMUdGJCIiIiwmKigtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSZhbHBoYUdGJEYoSSJ0R0YkRihGKC1JJGNvc0dGLTYjKiZJJWJldGFHRiRGKEYzRihGKEkiYUdGJEYoRigqKEYrRigtSSRzaW5HRi1GNkYoSSJiR0YkRighIiJGKEYoKiZJI2MyR0YkRigsJiooRitGKEY7RihGOUYoRigqKEYrRihGNEYoRj1GKEYoRihGKA==

Proceed to find a particular solution to the nonhomogeneous system using variation of parameters:

Step 4: Form the fundamental matrix X

PkkiWEc2Ii1JJDx8Z3I+R0koX3N5c2xpYkdGJDYkLCYqKC1JJGV4cEc2JCUqcHJvdGVjdGVkR0YnNiMqJkkmYWxwaGFHRiQiIiJJInRHRiRGMkYyLUkkY29zR0YtNiMqJkklYmV0YUdGJEYyRjNGMkYySSJhR0YkRjJGMiooRitGMi1JJHNpbkdGLUY2RjJJImJHRiRGMiEiIiwmKihGK0YyRjtGMkY5RjJGMiooRitGMkY0RjJGPUYyRjI=

Step 5: Compute its inverse, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjlRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIxRidGQUZBLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZB

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Step 6: Compute the product p = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjlRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIxRidGQUZBLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZB f (here f is the nonhomogeneous part of the differential equations right hand side)

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PkkicEc2Ii1JMGRlbGF5RG90UHJvZHVjdEc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiRJLFR5cGVzZXR0aW5nRzYkRihJKF9zeXNsaWJHRiQ2JEklaW52WEdGJEkiZkdGJA==

Step 7: Compute the integral of this product:

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Step 8: Form the particular solution xp as the product of X with this integral

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Step 9: The general solution xg to the nonhomogeneous system is then xg = xh + xp:

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Step 10: If we have initial conditions, solve for c1 and c2 to find solution to initial value problem.

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