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Multivariate Calculus

by: Gunner Price III

Multivariate Calculus MATH 250

Gunner Price III
Cal State Fullerton
GPA 3.92


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This 2 page Class Notes was uploaded by Gunner Price III on Wednesday September 30, 2015. The Class Notes belongs to MATH 250 at California State University - Fullerton taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/217029/math-250-california-state-university-fullerton in Mathematics (M) at California State University - Fullerton.

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Date Created: 09/30/15
Math 250B 63 The Method of Undetermined Coe eients Annihilators Summer 2007 Now we consider PDy FI 1 The general solution is yz yc1 m1 where yC is a solution to PDy 07 which we saw how to obtain in the last section Now we try to obtain the particular solution ypl Suppose that there is a polynomial differential operator AD such that ADF 0 Then apply AD to both sides of l to obtain AltDgtPltDgty 0 lt2 Any solution to l is a solution to The polynomial AD of lowest order that satis es 2 is called an annihilator of Fr7 and is said to annihilate Ft Strategy Find the general solution to 2 and choose the arbitrary constants to obtain a particular solution Example 1 Find the general solution to y 7 3y 2y 3651 Example 2 Show that AD D2 4 annihilates Fz 5 cos217 and nd the general solution to y 7 3y 2y 5 cos21l This method for obtaining a solution is called the method of undetermined coef cients The particular solution that we obtain as a general solution of 2 is called a trial solution This method applies only to equations in Which PD has constant coefficients and we can find an annihilator for Fl ANNlHlLATORS ll AD D 7 ak1 annihilates each of the functions and therefore it annihilates Fz a0 mm akzk ew7 for all constants a07 i i i 7aki Note that this applies in the special case When a Ol That is7 When Fz a0alzakzi E0 AD D2 7 2aD a2 b2 annihilates both of e cosbz and e sinbz7 and therefore it annihilates Fz 6 a0 cosbz be sinbz 7 for all constants a07 120 Note that When a 0 this applies to cosbz and sinbzi 3i AD D2 7 2aD a2 b2k1 annihilates each of the functions 6 cosbz7 z e cosbz7 I26 cosbz7 i i i 7 Ike cosbz e sinbz7 z e sinbz7 126 sinbz7 i i i 7 Ike sinbz and therefore it annihilates Fz a0 mm akzk e cosbz be 1211 b16116 sinbz7 for all constants a07 i i i 7ak7b07ui7bki F If Fz is a sum of functions of the forms in 137 then Fz is annihilated by the corresponding product of the annihilators in statements lSi Fz Usual Trial Solution Modified Trial Solution k If Pa 0 If a is a root of PU 0 of multiplicity m or 6 9271 Ao 39 39 39 Akrk 9171 Ime Ao Akrk lf Pa ib 0 If a 239 is a root of PU of multiplicity m e A0 cosbz Bo sinbz Ema A0 cosbz Bo sinbz czke cosbz IA1 cosbz B1 sinbz zA1 cosbz B1 sinbz or ypI 917I czke sinbz 1k Ak cosbz Bk sinbz zkAk cosbz Bk sinbz


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