Ordinary differential equation: Encyclopedia II - Ordinary differential equation - Types of differential equations with some history

Ordinary differential equation - Types of differential equations with some history

The influence of geometry, physics, and astronomy, starting with Newton and Leibniz, and further manifested through the Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert and Euler, has been very marked, and especially on the theory of linear partial differential equations with constant coefficients.

Ordinary differential equation - Homogeneous linear ODEs with constant coefficients

The first method of integrating linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form e^zx, for possibly-complex values of z. Thus

has the form

so dividing by e^zx gives the nth-order polynomial

In short the terms

of the original differential equation are replaced by z^k. Solving the polynomial gives n values of z, . Plugging those values into gives a basis for the solution; any linear combination of these basis functions will satisfy the differential equation.

This equation F(z) = 0, is the "characteristic" equation considered later by Monge and Cauchy.

has the characteristic equation

.

This has zeroes, i, −i, and 1 (multiplicity 2). The solution basis is then

, , , .

This corresponds to the real-valued solution basis

cosx, sinx, e^x, .

If z is a (possibly not real) zero of F(z) of multiplicity m and then is a solution of the ODE. These functions make up a basis of the ODE's solutions.

If the A_i are real then real-valued solutions are preferable. Since the non-real z values will come in conjugate pairs, so will their corresponding ys; replace each pair with their linear combinations Re(y) and Im(y).

A case that involves complex roots can be solved with the aid of Euler's formula.

• Example: Given . The characteristic equation is which has zeroes 2+i and 2−i. Thus the solution basis {y₁,y₂} is . Now y is a solution iff for .

Because the coefficients are real,

• we are likely not interested in the complex solutions
• our basis elements are mutual conjugates

The linear combinations

will give us a real basis in {u₁,u₂}.

Ordinary differential equation - Linear ODEs with constant coefficients

Suppose instead we face

For later convenience, define the characteristic polynomial

We find the solution basis as in the homogeneous (f=0) case. We now seek a particular solution y_p by the variation of parameters method. Let the coefficients of the linear combination be functions of x:

Using the "operator" notation D = d / dx and a broad-minded use of notation, the ODE in question is P(D)y = f; so

With the constraints

the parameters commute out, with a little "dirt":

But P(D)y_j = 0, therefore

This, with the constraints, gives a linear system in the u'_j. This much can always be solved; in fact, combining Cramer's rule with the Wronskian,

The rest is a matter of integrating u'_j.

The particular solution is not unique; also satisfies the ODE for any set of constants c_j.

See also variation of parameters.

Example: Suppose y'' − 4y' + 5 = sin(kx). We take the solution basis found above {e^{(2 + i)x},e^{(2 − i)x}}.

Using the list of integrals of exponential functions

And so

(Notice that u₁ and u₂ had factors that canceled y₁ and y₂; that is typical.)

For interest's sake, this ODE has a physical interpretation as a driven damped harmonic oscillator; y_p represents the steady state, and c₁y₁ + c₂y₂ is the transient.

Ordinary differential equation - Linear ODEs with variable coefficient

The method of undetermined coefficients (MoUC), is useful in finding solution for y_p. Given the ODE P(D)y = f(x), find another differential operator A(D) such that A(D)f(x) = 0. This operator is called the annihilator, and thus the method of undetermined coefficients is also known as the annihilator method. Applying A(D) to both sides of the ODE gives an homogeneous ODE for which we find a solution basis as before. Then the original nonhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combinations to satisfy the ODE.

Undetermined coefficients is not as general as variation of parameters in the sense that an annihilator does not always exist.

Example: Given y'' − 4y' + 5 = sin(kx), P(D) = D² − 4D + 5. The simplest annihilator of sin(kx) is A(D) = D² + k². The zeros of A(z)P(z) are {2 + i,2 − i,ik, − ik}, so the solution basis of A(D)P(D) is {y₁,y₂,y₃,y₄} = {e^{(2 + i)x},e^{(2 − i)x},e^ikx,e ^{− ikx}}.

Setting y = c₁y₁ + c₂y₂ + c₃y₃ + c₄y₄ we find

giving the system

which has solutions

giving the solution set

As explained above, the general solution to a non-homogeneous, linear differential equation y''(x) + p(x)y'(x) + q(x)y(x) = g(x) can be expressed as the sum of the general solution y_h(x) to the corresponding homogenous, linear differential equation y''(x) + p(x)y'(x) + q(x)y(x) = 0 and any one solution y_p(x) to y''(x) + p(x)y'(x) + q(x)y(x) = g(x).

Like the method of undetermined coefficients, described above, the method of variation of parameters is a method for finding one solution to y''(x) + p(x)y'(x) + q(x)y(x) = g(x), having already found the general solution to y''(x) + p(x)y'(x) + q(x)y(x) = 0. Unlike the method of undetermined coefficients, which fails except with certain specific forms of g(x), the method of variation of parameters will always work; however, it is significantly more difficult to use.

For a second-order equation, the method of variation of parameters makes use of the following fact:

Let p(x), q(x), and g(x) be functions, and let y₁(x) and y₂(x) be solutions to the homogeneous, linear differential equation y''(x) + p(x)y'(x) + q(x)y(x) = 0. Further, let u(x) and v(x) be functions such that u'(x)y₁(x) + v'(x)y₂(x) = 0 and u'(x)y₁'(x) + v'(x)y₂'(x) = g(x) for all x, and define y_p(x) = u(x)y₁(x) + v(x)y₂(x). Then y_p(x) is a solution to the non-homogeneous, linear differential equation y''(x) + p(x)y'(x) + q(x)y(x) = g(x).

y_p(x) = u(x)y₁(x) + v(x)y₂(x)

y_p''(x) + p(x)y'_p(x) + q(x)y_p(x) = g(x) + u(x)y₁''(x) + v(x)y₂''(x) + p(x)u(x)y₁'(x) + p(x)v(x)y₂'(x) + q(x)u(x)y₁(x) + q(x)v(x)y₂(x)

= g(x) + u(x)(y₁''(x) + p(x)y₁'(x) + q(x)y₁(x)) + v(x)(y₂''(x) + p(x)y₂'(x) + q(x)y₂(x)) = g(x) + 0 + 0 = g(x)

To solve the second-order, non-homogeneous, linear differential equation y''(x) + p(x)y'(x) + q(x)y(x) = g(x) using the method of variation of parameters, use the following steps:

1. Find the general solution to the corresponding homogeneous equation y''(x) + p(x)y'(x) + q(x)y(x) = 0. Specifically, find two linearly independent solutions y₁(x) and y₂(x).
2. Since y₁(x) and y₂(x) are linearly independent solutions, their Wronskian y₁(x)y₂'(x) − y₁'(x)y₂(x) is nonzero, so we can compute − (g(x)y₂(x)) / (y₁(x)y₂'(x) − y₁'(x)y₂(x)) and (g(x)y₁(x)) / (y₁(x)y₂'(x) − y₁'(x)y₂(x)). If the former is equal to u'(x) and the latter to v'(x), then u and v satisfy the two constraints given above: that u'(x)y₁(x) + v'(x)y₂(x) = 0 and that u'(x)y₁'(x) + v'(x)y₂'(x) = g(x). We can tell this after multiplying by the denominator and comparing coefficients.
3. Integrate − (g(x)y₂(x)) / (y₁(x)y₂'(x) − y₁'(x)y₂(x)) and (g(x)y₁(x)) / (y₁(x)y₂'(x) − y₁'(x)y₂(x)) to obtain u(x) and v(x), respectively. (Note that we only need one choice of u and v, so there is no need for constants of integration.)
4. Compute y_p(x) = u(x)y₁(x) + v(x)y₂(x). The function y_p is one solution of y''(x) + p(x)y'(x) + q(x)y(x) = g(x).
5. The general solution is c₁y₁(x) + c₂y₂(x) + y_p(x), where c₁ and c₂ are arbitrary constants.

The method of variation of parameters can also be used with higher-order equations. For example, if y₁(x), y₂(x), and y₃(x) are linearly independent solutions to y'''(x) + p(x)y''(x) + q(x)y'(x) + r(x)y(x) = 0, then there exist functions u(x), v(x), and w(x) such that u'(x)y₁(x) + v'(x)y₂(x) + w'(x)y₃(x) = 0, u'(x)y₁'(x) + v'(x)y₂'(x) + w'(x)y₃'(x) = 0, and u'(x)y₁''(x) + v'(x)y₂''(x) + w'(x)y₃''(x) = g(x). Having found such functions (by solving algebraically for u'(x), v'(x), and w'(x), then integrating each), we have y_p(x) = u(x)y₁(x) + v(x)y₂(x) + w(x)y₃(x), one solution to the equation y'''(x) + p(x)y''(x) + q(x)y'(x) + r(x)y(x) = g(x).

Solve the previous example, y'' + y = secx Recall . From technique learned from 3.1, LHS has root of that yield y_c = C₁cosx + C₂sinx, (so y₁ = cosx, y₂ = sinx ) and its derivatives

where the Wronskian

were computed in order to seek solution to its derivatives.

Upon integration,

Computing y_p and y_G:

Ordinary differential equation - General solution method for first-order linear ODEs

with the initial condition

.

Using the general solution method:

.

The integration is done from 0 to x, giving:

.

Then we can reduce to:

.

Assume that kappa is 2 from the initial condition.

For a first-order linear ODE, with coefficients that may or may not vary with t:

x'(t) + p(t)x(t) = r(t)

Then,

where κ is the constant of integration, and

This proof comes from Jean Bernoulli. Let

Suppose for some unknown functions u(t) and v(t) that x = uv.

Then

Substituting into the differential equation,

Now, the most important step: Since the differential equation is linear we can split this into two independent equations and write

Since v is not zero, the top equation becomes

The solution of this is

Substituting into the second equation

Since x = uv, for arbitrary constant C

As an illustrative example, consider a first order differential equation with constant coefficients:

This equation is particularly relevant to first order systems such as RC circuits, mass-damper systems.

After nondimensionalization, the equation becomes

In this case, p(t) = r(t) = 1.

Hence its solution by inspection is

Ordinary differential equation - Linear PDEs

The theory of linear partial differential equations may be said to begin with Lagrange (1779 to 1785). Monge (1809) treated ordinary and partial differential equations of the first and second order, uniting the theory to geometry, and introducing the notion of the "characteristic", the curve represented by F(z) = 0, which was investigated by Darboux, Levy, and Lie.

Ordinary differential equation - First-order PDEs

Pfaff (1814, 1815) gave the first general method of integrating partial differential equations of the first order, of which Gauss (1815) gave an analysis. Cauchy (1819) gave a simpler method, attacking the subject from the analytical standpoint, but using the Monge characteristic. Cauchy also first stated the theorem (now called the Cauchy-Kovalevskaya theorem) that every analytic differential equation defines an analytic function, expressible by means of a convergent series.

Jacobi (1827) also gave an analysis of Pfaff's method, besides developing an original one (1836) which Clebsch published (1862). Clebsch's own method appeared in 1866, and others are due to Boole (1859), Korkine (1869), and A. Mayer (1872). Pfaff's problem (on total differential equations) was investigated by Natani (1859), Clebsch (1861, 1862), DuBois-Reymond (1869), Cayley, Baltzer, Frobenius, Morera, Darboux, and Lie.

The next great improvement in the theory of partial differential equations of the first order was made by Lie (1872), who placed the whole subject on a solid foundation. After about 1870, Darboux, Kovalevsky, Méray, Mansion, Graindorge, and Imschenetsky became prominent in this line. The theory of partial differential equations of the second and higher orders, beginning with Laplace and Monge, was notably advanced by Ampère (1840).

The integration of partial differential equations with three or more variables was the object of elaborate investigations by Lagrange, and his name became connected with certain subsidiary equations. It was he and Charpit who originated one of the methods for integrating the general equation with two variables; a method which now bears Charpit's name.

Ordinary differential equation - Singular solutions

The theory of singular solutions of ordinary and partial differential equations was a subject of research from the time of Leibniz, but only since the middle of the nineteenth century did it receive special attention. A valuable but little-known work on the subject is that of Houtain (1854). Darboux (starting in 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field which was worked by various writers, notably Casorati and Cayley. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.

Ordinary differential equation - Reduction to quadratures

The primitive attempt in dealing with differential equations had in view a reduction to quadratures. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the nth degree, so it was the hope of analysts to find a general method for integrating any differential equation. Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless complex numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the first to appreciate the importance of this view. Thereafter the real question was to be, not whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or variables, and if so, what are the characteristic properties of this function.

Ordinary differential equation - The Fuchsian theory

Two memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor beginning in 1869, although his method for integrating a non-linear system was communicated to Bertrand in 1868. Clebsch (1873) attacked the theory along lines parallel to those followed in his theory of Abelian integrals. As the latter can be classified according to the properties of the fundamental curve which remains unchanged under a rational transformation, so Clebsch proposed to classify the transcendent functions defined by the differential equations according to the invariant properties of the corresponding surfaces f = 0 under rational one-to-one transformations.

Adapted from the Wikipedia article "Types of differential equations with some history", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki