<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

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<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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PkkjeGhHNiIsJiomSSNjMUdGJCIiIi1JJDwsPkdJKF9zeXNsaWJHRiQ2IywmKigtSSRleHBHNiQlKnByb3RlY3RlZEdGKzYjKiZJJmFscGhhR0YkRihJInRHRiRGKEYoLUkkY29zR0YxNiMqJkklYmV0YUdGJEYoRjZGKEYoSSJhR0YkRihGKCooRi9GKC1JJHNpbkdGMUY5RihJImJHRiRGKCEiIkYoRigqJkkjYzJHRiRGKC1GKjYjLCYqKEYvRihGPkYoRjxGKEYoKihGL0YoRjdGKEZARihGKEYoRig=

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

Step 4: Form the fundamental matrix X

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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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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIiVGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI3hnRidGL0YyRkE=