MATH24 Lecture 3 - MATH24 Differential Equations Equations...

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MATH24 Differential Equations
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Equations with Homogeneous Coefficients The differential equation M(x, y)dx + N(x, y)dy is homogeneous if both M and N are homogeneous and are of the same degree. Theorem: 1. If M(x, y) and N(x, y) are both homogeneous and of the same degree, the function or N/M is of degree 0. 2. If f(x, y) is homogeneous of degree zero in x + y, then f(x, y) is a function of y/x alone.
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General Solution of A Homogeneous D. E. If M(x, y) and N(x, y) are homogeneous and of degree zero, then the ratio M/N of N/M can be expressed as a function of single variable alone, say v. Thus, the substitution x = vy or y = vx will transform the equation to a variable separable D. E. Note: A differential equation is homogeneous if all the terms are of the same degree.
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Examples Test for homogeneity. 1. (x 2 + y 2 )dx + xydy = 0 M = x 2 + y 2 , homogeneous of degree 2 N = xy, homogeneous of degree 2 Thereforem, the D. E. is homogeneous. 2. (3x + 2y)dx – (x 2 + 2xy + y 2 )dy = 0 M = 3x + 2y, homogeneous of degree 1 N = x 2 + 2xy + y 2 , homogeneous of degree 2.
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This note was uploaded on 01/17/2012 for the course MATH 24 taught by Professor Dantesilva during the Spring '11 term at Mapúa Institute of Technology.

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MATH24 Lecture 3 - MATH24 Differential Equations Equations...

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