title: Homogeneous differential equation
text: # Homogeneous differential equation
  
  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
  
      $f(x,y)\,dy=g(x,y)\,dx,$
  
  where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form
  
      ${\frac {dx}{x}}=h(u)\,du,$
  
  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.
  
  ## History
  
  The term homogeneous was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article De integraionibus aequationum differentialium (On the integration of differential equations).
  
  ## Homogeneous first-order differential equations
  
  Differential equations :
  
  Scope :
  
  Fields
  
  Natural sciencesEngineering
  
  Astronomy
  
  Physics
  
  Chemistry
  
  Biology
  
  Geology
  
  Applied mathematics
  
  Continuum mechanics
  
  Chaos theory
  
  Dynamical systems
  
  Social sciences
  
  Economics
  
  Population dynamics
  
  List of named differential equations :
  
  Classification :
  
  Types
  
  Ordinary
  
  Partial
  
  Differential-algebraic
  
  Integro-differential
  
  Fractional
  
  Linear
  
  Non-linear
  
  By variable type
  
  Dependent and independent variables
  
  Autonomous
  
  Coupled / Decoupled
  
  Exact
  
  Homogeneous / Nonhomogeneous
  
  Features
  
  Order
  
  Operator
  
  Notation :
  
  Relation to processes
  
  Difference (discrete analogue)
  
  Stochastic
  
  Stochastic partial
  
  Delay :
  
  Solution :
  
  Existence and uniqueness
  
  Picard–Lindelöf theorem
  
  Peano existence theorem
  
  Carathéodory's existence theorem
  
  Cauchy–Kowalevski theorem :
  
  General topics
  
  Initial conditions
  
  Boundary values
  
  Dirichlet
  
  Neumann
  
  Robin
  
  Cauchy problem
  
  Wronskian
  
  Phase portrait
  
  Lyapunov / Asymptotic / Exponential stability
  
  Rate of convergence
  
  Series / Integral solutions
  
  Numerical integration
  
  Dirac delta function :
  
  Solution methods
  
  Inspection
  
  Method of characteristics
  
  Euler
  
  Exponential response formula
  
  Finite difference (Crank–Nicolson)
  
  Finite element
  
  Infinite element
  
  Finite volume
  
  Galerkin
  
  Petrov–Galerkin
  
  Green's function
  
  Integrating factor
  
  Integral transforms
  
  Perturbation theory
  
  Runge–Kutta
  
  Separation of variables
  
  Undetermined coefficients
  
  Variation of parameters :
  
  People :
  
  List
  
  Isaac Newton
  
  Gottfried Leibniz
  
  Jacob Bernoulli
  
  Leonhard Euler
  
  Józef Maria Hoene-Wroński
  
  Joseph Fourier
  
  Augustin-Louis Cauchy
  
  George Green
  
  Carl David Tolmé Runge
  
  Martin Kutta
  
  Rudolf Lipschitz
  
  Ernst Lindelöf
  
  Émile Picard
  
  Phyllis Nicolson
  
  John Crank :
  
  vte :
  
  A first-order ordinary differential equation in the form:
  
      $M(x,y)\,dx+N(x,y)\,dy=0$
  
  is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n. That is, multiplying each variable by a parameter λ, we find
  
      $M(\lambda x,\lambda y)=\lambda ^{n}M(x,y)\quad {\text{and}}\quad N(\lambda x,\lambda y)=\lambda ^{n}N(x,y)\,.$
  
  Thus,
  
      ${\frac {M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}}={\frac {M(x,y)}{N(x,y)}}\,.$
  
  ### Solution method
  
  In the quotient M(tx, ty)/N(tx, ty)=M(x, y)/N(x, y) {\textstyle {\frac {M(tx, ty)}{N(tx, ty)}}={\frac {M(x, y)}{N(x, y)}}}, we can let t = 1/x to simplify this quotient to a function f of the single variable y/x:
  
      ${\frac {M(x,y)}{N(x,y)}}={\frac {M(tx,ty)}{N(tx,ty)}}={\frac {M(1,y/x)}{N(1,y/x)}}=f(y/x)\,.$
  
  That is
  
      ${\frac {dy}{dx}}=-f(y/x).$
  
  Introduce the change of variables y = ux; differentiate using the product rule:
  
      ${\frac {dy}{dx}}={\frac {d(ux)}{dx}}=x{\frac {du}{dx}}+u{\frac {dx}{dx}}=x{\frac {du}{dx}}+u.$
  
  This transforms the original differential equation into the separable form
  
      $x{\frac {du}{dx}}=-f(u)-u,$
  
  or
  
      ${\frac {1}{x}}{\frac {dx}{du}}={\frac {-1}{f(u)+u}},$
  
  which can now be integrated directly: ln x equals the antiderivative of the right-hand side (see ordinary differential equation).
  
  ### Special case
  
  A first order differential equation of the form (a, b, c, e, f, g are all constants)
  
      $\left(ax+by+c\right)dx+\left(ex+fy+g\right)dy=0$
  
  where af ≠ be can be transformed into a homogeneous type by a linear transformation of both variables (α and β are constants):
  
      $t=x+\alpha ;\;\;z=y+\beta \,.$
  
  ## Homogeneous linear differential equations
  
  A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or any derivative of it. A linear differential equation that fails this condition is called inhomogeneous.
  
  A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is
  
      $L(y)=0$
  
  where L is a differential operator, a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function $f_i$ of x:
  
      $L=\sum _{i=0}^{n}f_{i}(x){\frac {d^{i}}{dx^{i}}}\,,$
  
  where $f_i$ may be constants, but not all $f_i$ may be zero.
  
  For example, the following linear differential equation is homogeneous:
  
      $\sin(x){\frac {d^{2}y}{dx^{2}}}+4{\frac {dy}{dx}}+y=0\,,$
  
  whereas the following two are inhomogeneous:
  
      $2x^{2}{\frac {d^{2}y}{dx^{2}}}+4x{\frac {dy}{dx}}+y=\cos(x)\,;$
      $2x^{2}{\frac {d^{2}y}{dx^{2}}}-3x{\frac {dy}{dx}}+y=2\,.$
  
  The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.
  
  