Non Homogeneous Differential Equation: A differential equation is a linear differential equation if it is expressible in the form P0 d^n y/dx^n + P1 d^n-1 y/dx^n-1 + P2 d^n-2 y/dx^n-2 + … + Pn-1 dy/dx + Pn y = Q, Where P0, P1, P2, …., Pn-1, Pn and Q are either constants or functions of independent variable x.Thus, if a differential equation when expressed in the form of a polynomial involves the derivatives and dependent variable in the first power and there are no product of these, and also the coefficient of the various terms are either constants or functions of the independent variable, then it is said to be linear differential equation. Otherwise, it is a non-differential equation.
Homogeneous differential equation: A function f(x, y) is called a homogeneous function of degree n if f(lambda x, lambda y) = lambda^n f(x, y).
For example, f(x, y) = x^2 – y^2 + 3xy is a homogeneous function degree 2, because f(lambda x, lambda y) = lambda^2 x^2 – lambda^2 y^2 + 3. Lambda x. lambda y = lambda^2 f(x, y).Let us understand Homogeneous Differential Equation Examples. Suppose we have to Solve x^2 y dx – (x^3 + y^3) dy = 0 . The given differential equation is x^2 y dx – (x^3 + y^3) dy = 0, dy/dx = (x^2 y)/(x^3 + y^3)……(i), Since each of the functions x^2 y and x^3 + y^3 is a homogeneous function of degree 3, so the given differential equation is homogeneous. Putting y = vx and dy/dx = v + x dv/dx in (i), we get v + x dv/dx = vx^3 /(x^3 + v^3 x^3), v + x dv/dx = v/1 + v^3, x dv/dx = [v/(1 + v^3)] – v, x dv/dx = (v – v – v^4)/(1 + v^3), x dv/dx = -v^4/(1 + v^3), x (1 + v^3) dv = -v^4 dx, (1 + v^3)/v^4 dv = -dx/x, (1/v^4 + 1/v) dv = -dx/x, Integrating both sides we get v^-3/-3 + log v = -log x + C=> -1/3v^3 + log v + log x = C=>-1/3 x^3/y^3 + log(y/x . x) = C=-1/3 . x^3/y^3 + log y = C, which is the required Homogeneous Solution Differential Equation.
Linear Homogeneous Differential Equation: A differential equation is linear if the dependent variable (y) and its derivative appear only in first degree. The general form of a linear differential equation of first order is dy/dx + Py = Q, where P and Q are functions of x (or constants).For example, dy/dx + xy = x^3, x dy/dx + 2y = x^3, dy/dx + 2y = sin x etc. are linear differential equations. Let us Find the General Solution of the Homogeneous Differential Equation (1 + y^2) dx + (1 + x^2) dy = 0. Now (1 + y^2) dx + (1 + x^2) dy = 0 => dx/(1 + x^2) + dy/(1 + y^2) = 0.On integration, we get tan^-1 x + tan^-1 y = tan^-1 C, (x + y)/(1 – xy) = C, X + y = C (1 – xy).
