# Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

## Notes

A function F(x, y) is said to be homogeneous function of degree n if F(λx, λy) = λ^n F(x, y) for any nonzero constant λ.
Consider the following functions in x and y:
1)F_1(x ,y) = y^2 + 2xy   ,
2) F_2(x,y) = 2x - 3y
3) F_3 (x, y) = sin x + cos y
If we replace x and y by λx and λy respectively in the above functions, for any nonzero constant λ, we get
1) F_1 (λx, λy) = λ^2 (y^2 + 2xy) = λ^2 F_1 (x, y)
2) F_2 (λx, λy) = λ (2x – 3y) = λ F_2 (x, y)
3) F_3 (λx, λy) = sin λx + cos λy ≠ λ^n F_3 (x, y), for any n ∈ N

Here, we observe that the functions F_1, F_2, can be written in the form F(λx, λy) = λ^n F (x, y) but F_3  can not be written in this form.
This leads to the above  definition.
We note that in the above examples,  F_1 , F_2 are homogeneous functions of degree 2, 1 respectively but F_3 is not a homogeneous function.
We also observe that

F_1(x,y) = x^2 ((y^2)/(x^2) + (2y)/x) = x^2h_1(y/x) or

F_1(x,y) = Y^2(1+(2x)/y) = y^2 h_2 (x/y)

F_2 (x,y) = x^1(2-(3y)/x) = x^1h_3(y/x)  or

F_2 (x,y) = y^1 (2x/y - 3) = y^1h_4(x/y)

F_3(x,y) ≠ x^nh_6 (y/x) , for any n ∈ N   or

F_3(x,y) ≠ y^nh_7 (x/y) , for any n ∈ N
Therefore, a function F (x, y) is a homogeneous function of degree n if
F(x,y) = x^n g (y/x)   or   y^nh(x/y)

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