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# 522 Review Sheet for M E 521 at PSU

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COURSE
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KARMA
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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15
Review Solutions of Ordinary Differential Equations Author John M Cimbala Penn State University Latest revision 06 September 2007 First Order Separable Ordinary Differential Equations Example Separate variables39 ydv Cdt C q I y yl y d I Solve a thq Example dy I I 7 C y yt 13 E r Separate variables ydy 7 d Solve 1222 cmg c139 First Order Linear Non Homogeneous Ordinary Differential Equations Example y yx gmxw Qx Multiply each term by an integrating factor Solve This leads to an exact differential on the LHS which can be solved easily vexp3Pac Higher Order Linear Homogeneous Ordinary Differential Equations with Constant Coef cients Example d3 d2 d d517 y yx d i 4 d Ey 6y 0 I Rewrite in terms ofa linear operator DnyE Here D34D2D6y0 I Factor the LHS Here D l D 2 D 3y 0 I Find the roots of the LHS Here the roots are l 2 and 3 I Solve yrgref1C 3932 3e34 Example 2y 2 I Rewrite in terms of the linear operator Here D2 a y 0 yyx 2 7 y0 I Factor the LHS Here Da Day0 I Find the roots of the LHS Here the roots are a and a Solve y ie 92 or 1 3 kt s M Example 2y 2 I Rewrite in terms of the linear operator Here D2 a y 0 y yx dXz a y 0 I Factor the LHS Here D ia D iay 0 Find the roots of the LHS Here the roots are ia and ia y CLei ax 6264 Solve or y 39IC3COSa 4rsz Sara Higher Order Nonlinear Non Homogeneous Ordinary Differential Equations Example y yx d3y dzy dy Pmyia 2 j Qx Define N unknowns where N is the highest order derivative of the original ecuation Here N 3 and define Split equation into N firstorder equations one for each F dF d3 d2 2 dF d2 dF 1 Qxpxf 2 3 dx dx dx dx dx dx dx Define derivative array in standard RungeKutta form in terms of the Nunknowns F1 F2 FN dFr 39sz ET Solve simultaneously for F1 F2 F Nusing the RungeKutta numerical technique and the boundary conditions D1 Qx1 iP395 Fi Ez 2 Di

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