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Homogeneous Ordinary Differential Equation


A linear ordinary differential equation of order n is said to be homogeneous if it is of the form

 a_n(x)y^((n))+a_(n-1)(x)y^((n-1))+...+a_1(x)y^'+a_0(x)y=0,
(1)

where y^'=dy/dx, i.e., if all the terms are proportional to a derivative of y (or y itself) and there is no term that contains a function of x alone.

However, there is also another entirely different meaning for a first-order ordinary differential equation. Such an equation is said to be homogeneous if it can be written in the form

 (dy)/(dx)=F(y/x).
(2)

Such equations can be solved in closed form by the change of variables u=y/x which transforms the equation into the separable equation

 (dx)/x=(du)/(F(u)-u).
(3)

See also

Homogeneous Function, Ordinary Differential Equation

Portions of this entry contributed by John Cook

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References

Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 8th ed. New York: Wiley, pp. 49-50, 2004.

Referenced on Wolfram|Alpha

Homogeneous Ordinary Differential Equation

Cite this as:

Cook, John and Weisstein, Eric W. "Homogeneous Ordinary Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HomogeneousOrdinaryDifferentialEquation.html

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