Homogeneous Linear Equations with Constant Coefficients

A second order homogeneous equation with constant coefficients is written as

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where a, b and c are constant. This type of equation is very useful in many applied problems (physics, electrical engineering, etc..). Let us summarize the steps to follow in order to find the general solution:

(1)
Write down the characteristic equation

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This is a quadratic equation. Let tex2html_wrap_inline80 and tex2html_wrap_inline82 be its roots tex2html_wrap_inline84 we have tex2html_wrap_inline86 tex2html_wrap_inline88 ;

(2)
If tex2html_wrap_inline80 and tex2html_wrap_inline82 are distinct real numbers (this happens if tex2html_wrap_inline94 ), then the general solution is

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(3)
If tex2html_wrap_inline96 (which happens if tex2html_wrap_inline98 ), then the general solution is

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(4)
If tex2html_wrap_inline80 and tex2html_wrap_inline82 are complex numbers (which happens if tex2html_wrap_inline104 ), then the general solution is

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where

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that is,

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Example: Find the solution to the IVP

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Solution: Let us follow the steps:

1
Characteristic equation and its roots

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Since 4-8 = -4<0, we have complex roots tex2html_wrap_inline114 . Therefore, tex2html_wrap_inline116 and tex2html_wrap_inline118 ;

2
General solution

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3
In order to find the particular solution we use the initial conditions to determine tex2html_wrap_inline122 and tex2html_wrap_inline124 . First, we have

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Since tex2html_wrap_inline128 , we get

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From these two equations we get

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which implies

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[Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Mohamed Amine Khamsi

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