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# Class Note for MATH 250A with Professor Lega at UA

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15
Calculus and Differential Equations ll MATH 250 B Other types of linear equations Cauchy Euler equations a Cauchy Euler equations are of the form ant y l an1t 1y 1 l l alty l aoy ft7 where the coefficients a are given constants o The change of variable t equot if t gt O and t iequot if t lt O transforms a Cauchy Euler equation into an equation with constant coefficients for y as a function of X The resulting equation may then be solved as usual 0 Of course the general solution yx must be re written in terms of t at the end of the calculation dzy dy dtz l HE l 25y cos 4 lnt l lnt 9 Example Solve t 2 for t gt 0 Higher order equations with constant coefficients 9 The methods discussed previously may easily be generalized o The characteristic equation associated with anyquot an71y 1 311 301 flx is a polynomial of order n with real coefficients 9 Because the coefficients are real the roots are either real or complex conjugate pairs 0 A real root A of multiplicity 1 gives a solution 5 O A real root A of multiplicity p gt 1 gives p solutions 5 X5 XZEAX I I I XpileAX 7 7 0 Complex conjugate roots A a i 5 give solutions 50 cos x and so sin x 0 Complex conjugate roots A a i B of multiplicity p gt1 give 2p solutions 50 cos x so sin x X 50 cos x X 50 sin x etc Higher order equations continued 0 The method of variation of parameters and the method of undetermined coefficients can also be extended 9 Example Solve ym 7 5yH12yli 8y e3t cos2t l Her 9 Variation of parameters for instance for a third order linear equation ify1y2 and y3 are three linearly independent solutions to the homogeneous equation we set Yp U1Y1 LIN2 U3Y3 With 0 UlY1U2 2U3Y37 0 uiyiu y u y 7 flxl 7 1Y1 2 2 3Y3 3X The determinant of this system which is the Wronskian of y1 y2 and y3 is non zero since the three solutions are linearly independent Other linear equations For linear equations of order n which do not have constant coefficients and are not Cauchy Euler equations try the following 0 Find by inspection one solution to the homogeneous equation 9 Use the method of reduction of order repeatedly to find n linearly independent solutions to the homogeneous equation 9 Write the general solution to the homogeneous equation yh as a linear combination of the above n linearly independent solutions 9 Find a particular solution yp by inspection by the method of undetermined coefficients or by variation of parameters 9 Write the general solution y yh yp 6 Apply initial or boundary conditions if any

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