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




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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