A differential equation can be homogeneous in either of two respects.
A first order differential equation is said to be homogeneous if it may be written
where f and g are homogeneous functions of the same degree of x and y.[1] In this case, the change of variable y = ux leads to an equation of the form
which is easy to solve by integration of the two members.
Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.
^Dennis G. Zill (15 March 2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning. ISBN 978-1-285-40110-2.
and 22 Related for: Homogeneous differential equation information
A differentialequation can be homogeneous in either of two respects. A first order differentialequation is said to be homogeneous if it may be written...
In mathematics, an ordinary differentialequation (ODE) is a differentialequation (DE) dependent on only a single independent variable. As with other...
the equation are partial derivatives. A linear differentialequation or a system of linear equations such that the associated homogeneousequations have...
solve inhomogeneous linear ordinary differentialequations. For first-order inhomogeneous linear differentialequations it is usually possible to find solutions...
In mathematics, a differentialequation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions...
In mathematics, a partial differentialequation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function...
mathematics and physics, the heat equation is a certain partial differentialequation. Solutions of the heat equation are sometimes known as caloric functions...
system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differentialequations) appear...
A stochastic differentialequation (SDE) is a differentialequation in which one or more of the terms is a stochastic process, resulting in a solution...
of a non-homogeneous linear ordinary differentialequation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary...
are exactly the solution of a specific partial differentialequation. More precisely: Euler's homogeneous function theorem — If f is a (partial) function...
for ordinary differentialequations are methods used to find numerical approximations to the solutions of ordinary differentialequations (ODEs). Their...
In mathematics, a system of differentialequations is a finite set of differentialequations. Such a system can be either linear or non-linear. Also, such...
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differentialequation that is quadratic in the unknown function...
In mathematics, an ordinary differentialequation is called a Bernoulli differentialequation if it is of the form y ′ + P ( x ) y = Q ( x ) y n , {\displaystyle...
A differentialequation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and...
tensor allows the EFE to be written as a set of nonlinear partial differentialequations when used in this way. The solutions of the EFE are the components...
it only works for differentialequations that follow certain forms. Consider a linear non-homogeneous ordinary differentialequation of the form ∑ i =...
P_{n}^{(\alpha ,\beta )}} is a solution of the second order linear homogeneousdifferentialequation ( 1 − x 2 ) y ″ + ( β − α − ( α + β + 2 ) x ) y ′ + n ( n...