Lesson 12a -Undetermined Coefficients

Thegoalwillbetofindoutwhichaandbwillmakeyp

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Unformatted text preview: d Coefficients Multiple Fun’s Say we had: ay ' 'by 'cy g ( x) f ( x) We simply find a particular solution for each function (one for g, and one for f). This will give us our overall solution as: y = yh+yp1+yp2 Example 1: Say we had: y ' '3 y '2 y Sin(2 x) Then we will first find the homogenous system: Auxiliary Equation is: r2+3r+2=0 Factoring/Quadratic Formula gives: r=‐1,‐2 This means our solution to the homogenous system is: yh c1e x c2 e 2 x y p ASin(2 x) BCos (2 x) Since is not part of our homogenous solution, we would try this one for our undetermined coefficients. The goal will be to find out which A and B will make yp a solution to the ODE. Example 1 Continued: y ' '3 y '2 y Sin(2 x) Sub in yp into: ( ASin(2 x) BCos (2 x))' '3( ASin(2 x) BCos (2 x))'2( ASin(2 x) BCos (2 x)) Sin(2 x) (2 ACos (2 x) 2 BSin(2 x))'3(2 ACos (2 x) 2 BSin(2 x...
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