# Lecture 10 - Non-Homogeneous Equations Method of...

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Non-Homogeneous Equations Method of Undetermined Coefficients

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We Know How To Solve Homogeneous Equations (With Constant Coefficients) Find Roots of Characteristic Polynomial A r 2 + B r + C = 0 Determine Appropriate General Solution y = C 1 e r 1 t + C 2 e r 2 t y = C 1 e r t + C 2 te r t y = B 1 e a t cos ( b t ) + B 2 e a t sin ( b t ) A y 00 + B y 0 + C y = 0
But what about Non-Homogeneous Equations? A y 00 + B y 0 + C y = g ( t ) Recall that we assumed the solution y = e r t For the homogeneous equation

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But what about Non-Homogeneous Equations? A y 00 + B y 0 + C y = g ( t ) For the Non-homogeneous equation, guess a different form of solution. Use g ( t ) as a guide
Example A y 00 + B y 0 + C y = g ( t ) 2 y 00 - 3 y 0 + y = 5 e 3 t

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Example 2 y 00 - 3 y 0 + y = 5 e 3 t Use 5 e 3 t to guess form of a solution suggests that e 3 t y = C e 3 t (This is the undetermined coefficient)
Example 2 y 00 - 3 y 0 + y = 5 e 3 t Use 5 e 3 t to guess form of a solution suggests that e 3 t y = C e 3 t Then: y 0 = 3 C e 3 t y 00 = 9 C e 3 t

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Example 2 y 00 - 3 y 0 + y = 5 e 3 t suggests that e 3 t y = C e 3 t Then: y 0 = 3 C e 3 t y 00 = 9 C e 3 t Plugging In: 2 3 5 - + y 00 y 0 y = e 3 t
Example 2 y 00 - 3 y 0 + y = 5 e 3 t suggests that e 3 t y = C e 3 t Then: y 0 = 3 C e 3 t y 00 = 9 C e 3 t Plugging In: 5 - + = e 3 t e 3 t C e 3 t C e 3 t C 9 18

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Example 2 y 00 - 3 y 0 + y = 5 e 3 t suggests that e 3 t y = C e 3 t Then: y 0 = 3 C e 3 t y 00 = 9 C e 3 t Plugging In: 5 = e 3 t e 3 t C 10
Example 2 y 00 - 3 y 0 + y = 5 e 3 t suggests that e 3 t y = C e 3 t Then: y 0 = 3 C e 3 t y 00 = 9 C e 3 t Plugging In: 5 = e 3 t e 3 t C 10 These are the same

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Example 2 y 00 - 3 y 0 + y = 5 e 3 t suggests that e 3 t y = C e 3 t Then: y 0 = 3 C e 3 t y 00 = 9 C e 3 t Plugging In: = C 1 2 Specific Solution: y = 1 2 e 3 t
Method of Undetermined Coefficients A y 00 + B y 0 + C y = g ( t ) Guess that specific solution takes the form: Use g ( t ) as a guide y = C f ( t ) (This is the undetermined coefficient)

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Method of
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