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firstord_undet

# firstord_undet - UNDETERMINED COEFFICIENTS for FIRST ORDER...

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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 ( ω t ), cos ( ω t ), 5. a sum of products e rt sin ( ω t ) , e rt cos ( ω t ), 6. a sum of terms p ( t ) , sin ( ω t ) + q ( t ) cos ( ω 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 = 8 , 2 A - 4 B = 0, and B - 4 C = 0, from which we see that A = - 2 , B = - 1, and C = - 1 / 4. Hence y p ( x ) = - 2 x 2 - x - 1 4 , and the general solution is 1

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y ( x ) = C e 4 x + - 2 x 2 - x - 1 4 . Example 2: Now consider the equation y - 4 y = 2 x cos ( x ). To find a particular solution, we take a
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firstord_undet - UNDETERMINED COEFFICIENTS for FIRST ORDER...

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