MATH24 Lecture 2 with correction

# MATH24 Lecture 2 with correction - MATH24 Differential...

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MATH24 Differential Equations

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Definitions Arbitrary Constants and Parameters An arbitrary constant is a constant that does not have a fixed value and is eliminated from an equation to form its differential equation while a parameter, usually treated as a constant, is not to be eliminated.
Steps in Eliminating Arbitrary Constants 1. Differentiate the equation as many times as the number of arbitrary constants. 2. Eliminate the arbitrary constants from the equation and its derivatives by a suitable method. Notes: 1. The desired differential equation is free from arbitrary constants. 2. The order of the resulting differential equation should be equal to the number of arbitrary constants.

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Examples Find the differential equations of the following by eliminating the arbitrary constants. 1. y = 3x 2 + ce −2x 2. y = C 1 sin4x + C 2 cos4x + x 3. y = (x + c)e −x 4. x + y = tan(x + c) Recall: tan −1 u = (du/dx)/(1 + u 2 ) 5. (x − a) 2 + y 2 = a 2 Families of Circles with center at (a, 0) and passing through the origin.
Families of Curves

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