# MATH 4B Lecture Notes - Lecture 8: Wronskian, Superposition Principle, Linear Algebra

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Second order linear homogenous ODE with constant coefficients:

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is a root of the characteristic equation

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By substitution, we've seen that

will be a solution

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: real distinct roots

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: complex conjugate roots

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What is the other solution that is linearly independent from ?

: repeated real roots

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The nature of the roots depends on the discriminant

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Characteristic equation:

Repeated real root

Find the general solution of

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Solutions:

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is called the variation of parameter

Substitute into

Is there a function so that

?

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For any constant ,

is a solution

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To check if

and

is a basis, compute the Wronskian

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Example:

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The general solution of , when is

,

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Summary:

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Repeated Roots

Second order linear homogeneous ODE:

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Then any linear combinations of and are also solutions

Superposition principle: Suppose that and are solutions of the second order

linear homogenous ODE

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If and are solutions whose Wronskian , then

represent a

fundamental set of solutions

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The general solution of the homogenous equation above is

for

arbitrary constants

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Review of Homogeneous Case

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Second order linear inhomogeneous ODE:

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Suppose that is some (particular) solution of the inhomogeneous equation

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Suppose the is the general solution of the homogeneous equation

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Inhomogeneous Case

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The Superposition Principle for Inhomogeneous Equations

Lecture 8

Tuesday, February 7, 2017

5:34 PM

MATH 4B Page 1