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HomoEquations

# HomoEquations - Homogeneous Equations A homogeneous...

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Homogeneous Equations A homogeneous equation can be transformed into a separable equation by a change of variables. Definition : An equation in differential form M(x,y) dx + N(x,y) dy = 0 is said to be homogeneous , if when written in derivative form dy dx = f(x,y) = g y x ! " # \$ % & there exists a function g such that f(x,y) = g y x ! " # \$ % & . Example : (x 2 – 3y 2 ) dx – xy dy = 0 is homogeneous since dy dx = x 2 ! 3y 2 xy = x y ! 3 y x = y x " # \$ % & ! 1 ! 3 y x = g y x " # \$ % & . We can use another approach to define a homogeneous equation. Definition : A function F(x,y) of the variables x and y is called homogeneous of degree n if for any parameter t F(tx, ty) = t n F(x,y) Example : Given F(x,y) = x 3 – 4x 2 y + y 3 , it is a homogeneous function of degree 3 since F(tx,ty) = (tx) 3 – 4(tx) 2 (ty) + (ty) 3 = t 3 (x 3 – 4x 2 y + y 3 ). Theorem : The O.D.E. in differential form M(x,y) dx + N(x,y) = 0 is a homogeneous O.D.E. if M(x,y) and N(x,y) are homogeneous functions of the same degree. Proof : Assume M(x,y) and N(x,y) are homogeneous functions of degree n, then M(tx,ty) = t n M(x,y) and N(x,y) = t n

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