section%202.5 - Chapter 2 First Order Differential...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 1 ...
View Full Document

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.

Ask a homework question - tutors are online