Homogeneous Form

•    In order to understand homogeneous differential equations, it is important to understand homogeneous function. A function f(x, y) is called a homogeneous function of degree n, if a function f can be expressed as follows:

•    Homogeneous Differential Equation: A differential equation of the form dy/dx = F(x, y) is said to be homogeneous differential equation if the function F(x, y) is a homogeneous function of degree zero.

•    Solution of Homogeneous Differential Equation:
o    Substitute y = vx and dy/dx = v + x (dv/dx) in the differential equation.
o    Separate the variables and integrate on both sides.
o    Replace v by y/x.

•    Differential Equation Reducible to Homogeneous Form:
•    A differential equation of the form dy/dx = [a1x + b1y + c1]/ [a2x + b2y + c2] can be reduced to homogeneous form by using the following transformations:
x = X + h and y =Y + k, where, h and k are constants to be determined so that c1 and c2 are eliminated.

•    There are various real life usages of differential equations. An ideal spring with a spring constant k is described by the simple harmonic oscillation. Newton's second law gives its equation of motion in the form of a homogeneous second-order linear differential equation.

Keywords: Homogeneous differential equation, Homogeneous function