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Unformatted text preview: Method of undetermined coefficients Igors Gorbovickis November 15, 2009 The method of undetermined coefficients helps to find a particular so- lution of a linear nonhomogeneous differential equation provided that the equation satisfies some certain conditions. Definition 1. A particular solution is just any solution of a given differential equation. When applicable? When the equation has constant coefficients (the coefficients do not depend on x ), and the right part of the equation is the sum of products of polynomials and functions of the form e ax , cos( bx ) and sin( cx ) . Example 2. The method is applicable to the equation 4 y 00 + y- e y = x + 2 sin 2 (3 x ) + x 2 e 6 x cos(2 x ) sin(7 x ) , and is not applicable to the equations y 00 + xy- y = cos( x ) and y 00- 2 y + 2 y = e sin x . The method Step 1: Solving the corresponding homogeneous equation We solve the same equation but with the zero right part (the corresponding homogeneous equation). That equation can be solved by using the charac- teristic equation. The general solution of the homogeneous equation is called the complementary solution of the original equation. 1 Example 3. y 00 + 4 y = 0 . The characteristic equation is r 2 + 4 = 0 . The roots are r = 2 i , and y c = A cos(2 x ) + B sin(2 x ) . Remark 4. We can obtain a general solution of a linear nonhomogeneous equation by taking its complementary solution and adding any particular solution to it. Thus the method of undetermined coefficients actually gives us all solutions of the original equation....
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