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Unformatted text preview: UNDETERMINED COEFFICIENTS for FIRST ORDER LINEAR EQUATIONS This method is useful for solving non-homogeneous linear equations written in the form dy dx + k y = g ( x ) , where k is a non-zero constant and g is 1. a polynomial, 2. an exponential e rt , 3. a product of an exponential and a polynomial, 4. a sum of trigonometric functions sin ( t ), cos ( t ), 5. a sum of products e rt sin ( t ) , e rt cos ( t ), 6. a sum of terms p ( t ) , sin ( t ) + q ( t ) cos ( t ), where p and q are polynomials. Here are a couple more examples. Example 1: Find the general solution of y- 4 y = 8 x 2 . Here we take a trial solution to be a general polynomial of degree two y p ( x ) = A x 2 + B x + C . Then y p ( x ) = 2 Ax + B and substituting we have (2 A x + B )- 4 ( A x 2 + B x + C ) = 8 x 2 . Now, collecting like powers of x we rewrite this equation as- 4 A x 2 + (2 A- 4 B ) x + ( B- 4 C ) = 8 x 2 , and comparing coefficients of like terms on both sides of the equation gives- 4 A =...
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