2.1 Linear - Integrating Factors

2.1 Linear - Integrating Factors - GE 207K Solving...

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GE 207K Solving First-Order Linear ODE’s September 1, 2011 Once a diferential equation has been identiFed as linear and first order ,i . e . th e diferential equation can be written in the the ±orm y ° + p ( t ) y = q ( t ) , the ±ollowing steps can be taken to Fnd the general solution to the diferential equation: Steps: 1. Write the given diferential equation in the STANDARD FORM y ° + p ( t ) y = q ( t ) . (1) 2. ²ind the integrating factor , μ : μ ( t )= e R p ( t ) dt . (2) 3. Multiply the BOTH sides o± the Eq. ( 2 )bytheintegrat ing±actor : e R p ( t ) dt [ y ° + p ( t ) y ]= q ( t ) e R p ( t ) dt . (3) 4. Note that the l e±t-h and-s ide (LHS) o± Eq. ( 3 ) can now be re-written as d dt ° ye R p ( t ) dt ± = q ( t ) e R p ( t ) dt . (4) 5. Integrate both sides o± Eq. ( 4 ). Don’t ±orget to include the integration constant! ye R p ( t ) dt = ² q ( t ) e R p ( t ) dt dt + C. (5) 6. Move the exponential to the r ight-h and-s ide (RHS) to get the solution in terms o± y : y = e R p ( t ) dt ³² q ( t ) e R p ( t ) dt dt + C ´ .
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GE 207K Solving First-Order Linear ODE’s September 1, 2011 Examples solved in class: Example 1 y ° y = e 2 x Step 1: The diferential equation is already in standard Form. p ( t )= 1 . Step 2: ±ind the integrating Factor: μ = e R 1 dx = e x . Step 3: Multiply both sides oF the diferential equation by μ . e x [ y ° y ]= e x e 2 x . (7) Step 4: Rewrite Eq. ( 7 )as Notation: ` e x y ´ ° d dx ` e x y ´ ° e x y ± ° = e x . (8) Step 5: Integrate both sides oF Eq. ( 8 ): e x y = e x + C. Step 6: Rearrange and separate y on LHS: y = e 2 x + Ce x . Note: You may choose to skip Step 3 and go to Step 4 directly. But when you skip it, don’t ! Forget to multiply the RHS oF equation by the integrating Factor! ±orgetting to multiply the RHS by the integrating Factor is a very common mistake !
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This note was uploaded on 02/03/2012 for the course M 427K taught by Professor Fonken during the Fall '08 term at University of Texas at Austin.

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2.1 Linear - Integrating Factors - GE 207K Solving...

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