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clase10handouts - Euler-Lagrange method for solving linear...

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Unformatted text preview: Euler-Lagrange method for solving linear first-order differential equations To solve a linear DE y + a ( t ) y = f ( t ) , where a and f are continuous on a domain I , take the following steps: 1. Solve y + a ( t ) y = 0 by separation of variables to obtain y h = Ce- R a ( t ) dt , C arbitrary constant . 2. Solve v ( t ) e- R a ( t ) dt = f ( t ) for v ( t ) to obtain a particular solution y p = v ( t ) e- R a ( t ) dt . M.I. Bueno Differential Equations and Linear Algebra Euler-Lagrange method for solving linear first-order differential equations 3. Combine the results of Steps 1 and 2 to form the general solution y ( t ) = y h + y p . 4. If you are solving an IVP, only after Step 3 can you substitute the initial condition to find C . M.I. Bueno Differential Equations and Linear Algebra Integrating factor method for first-order linear DEs To solve a linear DE y + a ( t ) y = f ( t ) , where a and f are continuous on a domain I : 1. Find the integrating factor μ ( t ) = e R a ( t ) dt , and pick the arbitrary constant in the antiderivative to be zero. Note that μ ( t ) 6 = 0 for t ∈ I . 2. Multiply each side of the differential equation by the integrating factor to get e R a ( t ) dt [ y + a ( t ) y ] = f ( t ) e R a ( t ) dt , which reduces to d dt h e R a ( t ) dt y ( t ) i = f ( t ) e R a ( t ) dt . M.I. Bueno Differential Equations and Linear Algebra Integrating factor method for first-order linear DEs 3. Find the antiderivative of the final equation in Step 2 to get e R a ( t ) dt y ( t ) = Z f ( t ) e R a ( t ) dt dt + C ....
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This note was uploaded on 03/14/2010 for the course MATH 3C taught by Professor Jacobs during the Fall '08 term at UCSB.

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clase10handouts - Euler-Lagrange method for solving linear...

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