Английская Википедия:Homogeneous differential equation
Шаблон:Short description 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
- <math>f(x,y) \, dy = g(x,y) \, dx,</math>
where Шаблон:Mvar and Шаблон:Mvar are homogeneous functions of the same degree of Шаблон:Mvar and Шаблон:Mvar.[1] In this case, the change of variable Шаблон:Math leads to an equation of the form
- <math>\frac{dx}{x} = h(u) \, du,</math>
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).[2]
Homogeneous first-order differential equations
A first-order ordinary differential equation in the form:
- <math>M(x,y)\,dx + N(x,y)\,dy = 0 </math>
is a homogeneous type if both functions Шаблон:Math and Шаблон:Math are homogeneous functions of the same degree Шаблон:Mvar.[3] That is, multiplying each variable by a parameter Шаблон:Math, we find
- <math>M(\lambda x, \lambda y) = \lambda^n M(x,y) \quad \text{and} \quad N(\lambda x, \lambda y) = \lambda^n N(x,y)\,. </math>
Thus,
- <math>\frac{M(\lambda x, \lambda y)}{N(\lambda x, \lambda y)} = \frac{M(x,y)}{N(x,y)}\,. </math>
Solution method
In the quotient <math display="inline">\frac{M(tx,ty)}{N(tx,ty)} = \frac{M(x,y)}{N(x,y)}</math>, we can let Шаблон:Math to simplify this quotient to a function Шаблон:Mvar of the single variable Шаблон:Math:
- <math>\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)\,. </math>
That is
- <math>\frac{dy}{dx} = -f(y/x).</math>
Introduce the change of variables Шаблон:Math; differentiate using the product rule:
- <math>\frac{dy}{dx}=\frac{d(ux)}{dx} = x\frac{du}{dx} + u\frac{dx}{dx} = x\frac{du}{dx} + u.</math>
This transforms the original differential equation into the separable form
- <math>x\frac{du}{dx} = -f(u) - u, </math>
or
- <math>\frac 1x\frac{dx}{du} = \frac {-1}{f(u) + u}, </math>
which can now be integrated directly: Шаблон:Math equals the antiderivative of the right-hand side (see ordinary differential equation).
Special case
A first order differential equation of the form (Шаблон:Mvar, Шаблон:Mvar, Шаблон:Mvar, Шаблон:Mvar, Шаблон:Mvar, Шаблон:Mvar are all constants)
- <math> \left(ax + by + c\right) dx + \left(ex + fy + g\right) dy = 0</math>
where Шаблон:Math can be transformed into a homogeneous type by a linear transformation of both variables (Шаблон:Mvar and Шаблон:Mvar are constants):
- <math>t = x + \alpha; \;\; z = y + \beta \,. </math>
Homogeneous linear differential equations
Шаблон:See also A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if Шаблон:Math is a solution, so is Шаблон:Math, for any (non-zero) constant Шаблон:Mvar. 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 Шаблон:Math where Шаблон:Mvar is usually the independent variable and Шаблон:Mvar is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is
- <math> L(y) = 0</math>
where Шаблон:Mvar is a differential operator, a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function Шаблон:Math of Шаблон:Mvar:
- <math> L = \sum_{i=0}^n f_i(x)\frac{d^i}{dx^i} \, ,</math>
where Шаблон:Math may be constants, but not all Шаблон:Math may be zero.
For example, the following linear differential equation is homogeneous:
- <math> \sin(x) \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + y = 0 \,, </math>
whereas the following two are inhomogeneous:
- <math> 2 x^2 \frac{d^2y}{dx^2} + 4 x \frac{dy}{dx} + y = \cos(x) \,; </math>
- <math> 2 x^2 \frac{d^2y}{dx^2} - 3 x \frac{dy}{dx} + y = 2 \,. </math>
The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.
See also
Notes
References
- Шаблон:Citation. (This is a good introductory reference on differential equations.)
- Шаблон:Citation. (This is a classic reference on ODEs, first published in 1926.)
- Шаблон:Cite book
- Шаблон:Cite book
External links
- Homogeneous differential equations at MathWorld
- Wikibooks: Ordinary Differential Equations/Substitution 1
Шаблон:Differential equations topics
