Unformatted text preview: Chapter 2. First Order Diﬀerential Equations §2.5 Solutions by Substitutions 1. Homogeneous functions and homogeneous equations: If a function f (x, y ) possesses the property f (tx, ty ) = tα f (x, y ) for some real number α, then f is said to be a homogeneous function of degree α. A ﬁrst order diﬀerential equation of the form M (x, y )dx + N (x, y )dy = 0 is said to be a homogeneous if both M and N are homogeneous of the same degree. To solve an homogeneous equation, use the substitution y = ux or x = vy to reduce the equation to a separable DE. EX: Solve the following DEs. (x2 + y 2 )dx + (x2 − xy )dy = 0 2. Bernoulli’s Equation: The equation dy + P (x) = f (x)y n dx where n is any real number, is called Bernoulli’s Equation. To slove this equation, use substitution u = y 1−n . EX: Solve x dy + y = x2 y 2 . dx 3. Other Substitutions: A DE in the form of dy = f (Ax + By + C ) dx can always be reduced to an separable equation by the substitution u = Ax + By + C. EX: Solve dy = (x + y + 1)2 dx
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This note was uploaded on 01/05/2011 for the course MATH 3310 taught by Professor Dr.du during the Fall '08 term at Kennesaw.
 Fall '08
 DR.DU
 Differential Equations, Equations

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