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Homogeneous Equations

The function f(x, y) is said to be a homogeneous function of degree n if for any real number t(≠ 0), we have f(tx, ty) = tn f(x, y), e.g., f(x, y) = ax2/3 + hx1/3 y1/3 + by2/3 is a homogeneous function of degree 2/3.
A differential equation of the form 102217.png = 102211.png, where f(x, y) and φ (x, y) are homogeneous functions of x and y, and of the same degree, is called homogeneous.
This equation may also be reduced to the form 102205.png = 102199.png and is solved by putting y = vx, so that the dependent variable y is changed to another variable v, where v is some unknown function. The differential equation is transformed to an equation with variables separable.

Equations reducible to the homogeneous form

Equations of the form 102187.png (aBAb and A + b ≠ 0) can be reduced to a homogeneous form by changing the variable x, y, to X, Y by writing x = X + h and y = Y + k; where h, k are constants to be chosen so as to make the given equation homogeneous. We have 102181.png. Hence the given equation becomes 102174.png.
Let h and k be so chosen as to satisfy the relation ah + bk + c = 0 and Ah + Bk + C = 0. These give
h = 102168.png, which are meaningful when aBAb. 102162.png can now be solved by means of the substitution Y = VX.
In case aB = Ab, we write ax + by = t. This reduces the differential equation to the separable variable type.
If A + b = 0 then a simple cross-multiplication and substituting d(xy) for xdx + ydy and integrating term by term yields the result easily.

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