Lecture 11 - Variation of Parameters Method for...

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Variation of Parameters Method for Non-Homogeneous Equations
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Recall Non-Homogeneous Linear Equation A y 00 + B y 0 + C y = g ( t ) Solve by finding specific solution And finding general solution to the Homogeneous Equation y y h A y 00 + B y 0 + C y = 0 The General Solution takes the form: y + y h
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Last Time We found specific solutions to the Non-Homogeneous Equation y Using Undetermined Coefficients: Guess that specific solution takes the form: y = C f ( t ) Plug in to differential equation Solve for C
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Undetermined Coefficients is Only Appropriate for Certain g(t) g ( t ) e t cos ( β t ) or sin ( β t ) e t cos ( β t ) or e t sin ( β t ) a n t n + a n - 1 t n - 1 + ... + a 0 Times anything above
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Undetermined Coefficients is Only Appropriate for Certain g(t) What if g ( t ) doesn’t take that form? Use a more general technique Variation of Parameters
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Variation of Parameters Suppose we know the general solution to the Homogeneous Equation A y 00 + B y 0 + C y = g ( t ) A y 00 + B y 0 + C y = 0 Which takes the form y = C 1 C 2 y 1 y 2 + Search for a specific solution of the form y = y 1 y 2 + u 1 ( t ) u 2 ( t )
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Variation of Parameters A y 00 + B y 0 + C y = g ( t ) Search for a specific solution of the form y = y 1 y 2 + u 1 ( t ) u 2 ( t ) Again, plug in to the equation, find conditions on u 1 ( t ) u 2 ( t ) y 0 = y 0 1 u 1 ( t ) u 0 1 ( t ) u 0 2 ( t ) y 0 2 u 2 ( t ) y 1 y 2 + + + So: y 00 = So… many… terms…
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An Important Aside Linear Algebra Question: 3 x 1 + 7 x 2 = 5 How many solutions? (One equation, Two unknowns…) Infinite!
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An Important Aside Linear Algebra Question: 3 x 1 + 7 x 2 = 5 How many solutions? Just one. But if we add one condition 1 x 1 + - 1 x 2 = 0
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An Important Aside This holds for Must satisfy (One equation, two unknowns) y = y 1 y 2 + u 1 ( t ) u 2 ( t ) A y 00 + B y 0 + C y = g ( t ) Infinite different u 1 ( t ) u 2 ( t ) Will satisfy the equation
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An Important Aside This holds for y = y 1 y 2 + u 1 ( t ) u 2 ( t ) Infinite different u 1 ( t ) u 2 ( t ) Will satisfy the equation Let’s add a condition = 0 Now, u 1 ( t ) u 2 ( t ) Are uniquely defined u 0 1 ( t ) y 1 u 0 2 ( t ) y 2 +
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Variation of Parameters A y 00 + B y 0 + C y = g ( t ) Search for a specific solution of the form y = y 1
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