Given an Example of a Family of Differential

2007 Schools Wikipedia Selection. Related subjects: Mathematics

In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one contained variable, and one or more of its derivatives with respect to that variable.

A simple example is Newton's second law of motion, which leads to the differential equation

$m \frac{d^2 x}{dt^2} = f(x)\,$ ,

for the motion of a particle of mass grand. In full general, the force f depends upon the position of the particle 10, and thus the unknown variable x appears on both sides of the differential equation, as is indicated in the notation f(ten).

Ordinary differential equations are to be distinguished from partial differential equations where in that location are several independent variables involving partial derivatives.

Ordinary differential equations arise in many dissimilar contexts including geometry, mechanics, astronomy and population modelling. Many famous mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoullis, Riccati, Clairaut, d'Alembert and Euler.

Much report has been devoted to the solution of ordinary differential equations. In the case where the equation is linear, it can be solved by analytical methods. Unfortunately, nigh of the interesting differential equations are non-linear and, with a few exceptions, cannot be solved exactly. Approximate solutions are arrived at using computer approximations (come across numerical ordinary differential equations).

Definitions

Let y be an unknown function

$y: \mathbb{R} \to \mathbb{R}$

in x with $y (i)$ the i-th derivative of y, so a function

$F(x,y,y^{(1)},\ \dots,\ y^{(n-1)})=y^{(n)}$

is called an ordinary differential equation (ODE) of club (or degree) n. For vector valued functions

$y: \mathbb{R} \to \mathbb{R}^m$

we telephone call F a system of ordinary differential equations of dimension yard.

A office y is called a solution of F. A general solution of an northwardth-order equation is a solution containing $northward$ arbitrary variables, corresponding to due north constants of integration. A particular solution is derived from the full general solution by setting the constants to particular values. A singular solution is a solution that can't be derived from the full general solution.

When a differential equation of society due north has the form

$F\left(x, y, y', y'',\ \dots,\ y^{(n)}\right) = 0$

it is chosen an implicit differential equation whereas the class

$F\left(x, y, y', y'',\ \dots,\ y^{(n-1)}\right) = y^{(n)}$

is called an explicit differential equation.

A differential equation not depending on x is chosen autonomous.

A differential equation is said to be linear if F can exist written equally a linear combination of the derivatives of y

$y^{(n)} = \sum_{i=1}^{n-1} a_i(x) y^{(i)} + r(x)$

with a _i(x) and r(10) continuous functions in x. If r(x)=0 the we call the linear differential equation homogeneous otherwise we call it inhomogeneous.

Examples

Reduction of dimension

Given an explicit ordinary differential equation of order due north and dimension 1,

$F\left(x, y, y', y'',\ \dots,\ y^{(n-1)}\right) = y^{(n)}$

we define a new family of unknown functions

y n : = y (n - ane) .

We can and so rewrite the original differential equation as a system of differential equations with social club one and dimension n.

$y_1^' = y_2$

$\vdots$

$y_n^' = F(y_n, \dots, y_1, x).$

which tin be written concisely in vector note as

$\mathbf{y}^'=\mathbf{F}(\mathbf{y}, x)$

with

$\mathbf{y}:=(y,\ldots,y_n).$

Types of ordinary differential equations

Ordinary differential equations which tin be categorised past three factors:

Linear vs. Non-linear
Homogeneous vs. Inhomogenous
Abiding coefficents versus variable coefficients

Information below provides methods for the solution of these differing ODEs:

Homogeneous linear ODEs with constant coefficients

The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions accept the grade $eastward z ten$ , for possibly-complex values of $z$ . Thus to solve

$\frac {d^{n}y} {dx^{n}} + A_{1}\frac {d^{n-1}y} {dx^{n-1}} + \cdots + A_{n}y = 0$

we set $y = e z x$ , leading to

$z^n e^{zx} + A_1 z^{n-1} e^{zx} + \cdots + A_n e^{zx} = 0$

then dividing by $e z 10$ gives the northwardth-social club polynomial

$F(z) = z^{n} + A_{1}z^{n-1} + \cdots + A_n = 0$

In short the terms

$\frac {d^{k}y} {dx^{k}}\quad\quad(k = 1, 2, \cdots, n).$

of the original differential equation are replaced by z ^grand. Solving the polynomial gives n values of z, $z_1, \dots,z_n$ . Plugging those values into $e^{z_i x}$ gives a footing for the solution; any linear combination of these footing functions volition satisfy the differential equation.

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

Example

$y''''-2y'''+2y''-2y'+y=0 \,$

has the characteristic equation

$z^4-2z^3+2z^2-2z+1=0 \,$ .

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

$e^{ix} ,\, e^{-ix} ,\, e^x ,\, xe^x \,.$

This corresponds to the real-valued solution basis

$\cos x ,\, \sin x ,\, e^x ,\, xe^x \,.$

If z is a (perchance non real) zilch of F(z) of multiplicity thou and $k\in\{0,1,\dots,m-1\} \,$ and then $y=x^ke^{zx} \,$ is a solution of the ODE. These functions make upward a basis of the ODE's solutions.

If the A_i are real and then real-valued solutions are preferable. Since the non-real z values will come in conjugate pairs, so volition their corresponding ydue south; supercede 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.

Considering the coefficients are real,

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

The linear combinations

$u_1=\mbox{Re}(y_1)=\frac{y_1+y_2}{2}=e^{2x}\cos(x) \,$ and

$u_2=\mbox{Im}(y_1)=\frac{y_1-y_2}{2i}=e^{2x}\sin(x) \,$

will give usa a real footing in ${u i, u 2}$ .

Inhomogeneous linear ODEs with constant coefficients

Suppose instead we confront

$\frac {d^{n}y} {dx^{n}} + A_{1}\frac {d^{n-1}y} {dx^{n-1}} + \cdots + A_{n}y = f(x).$

For later convenience, ascertain the feature polynomial

$P(v)=v^n+A_1v^{n-1}+\cdots+A_n.$

We discover the solution basis $\{y_1,y_2,\ldots,y_n\}$ every bit in the homogeneous (f=0) instance. We now seek a detail solution y_p by the variation of parameters method. Let the coefficients of the linear combination be functions of x:

$y_p=u_1y_1+u_2y_2+\cdots+u_ny_n.$

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

$f=P(D)y_p=P(D)(u_1y_1)+P(D)(u_2y_2)+\cdots+P(D)(u_ny_n).$

With the constraints

$0=u'_1y_1+u'_2y_2+\cdots+u'_ny_n$

$0=u'_1y'_1+u'_2y'_2+\cdots+u'_ny'_n$

…

$0=u'_1y^{(n-2)}_1+u'_2y^{(n-2)}_2+\cdots+u'_ny^{(n-2)}_n$

the parameters commute out, with a fiddling "clay":

$f=u_1P(D)y_1+u_2P(D)y_2+\cdots+u_nP(D)y_n+u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+\cdots+u'_ny^{(n-1)}_n.$

Just $P (D) y j = 0$ , therefore

$f=u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+\cdots+u'_ny^{(n-1)}_n.$

This, with the constraints, gives a linear arrangement in the $u' j$ . This much tin always be solved; in fact, combining Cramer'due south rule with the Wronskian,

$u'_j=(-1)^{n+j}\frac{W(y_1,\ldots,y_{j-1},y_{j+1}\ldots,y_n)_{0 \choose f}}{W(y_1,y_2,\ldots,y_n)}.$

The balance is a affair of integrating $u' j .$

The particular solution is not unique; $y_p+c_1y_1+\cdots+c_ny_n$ also satisfies the ODE for whatsoever set of constants c_j .

Meet also variation of parameters.

Instance: Suppose $y'' - four y' + 5 y = due south i n (g x)$ . Nosotros take the solution ground constitute in a higher place ${due east (two + i) x, e (2 - i) x}$ .

Using the list of integrals of exponential functions

And then

(Notice that u ₁ and u ₂ had factors that canceled y ₁ and y _two; that is typical.)

For interest'south sake, this ODE has a concrete interpretation as a driven damped harmonic oscillator; y_p represents the steady state, and $c 1 y 1 + c 2 y 2$ is the transient.

Commencement-club linear ODEs

Example

$y'+3y=2 \,$

with the initial condition

$f\left(0\right)=2 \,$ .

Using the general solution method:

$f=e^{-3x}\left(\int 2 e^{3x} dx + \kappa\right) \,$ .

The integration is done from 0 to x, giving:

$f=e^{-3x}\left(2/3\left( e^{3x}-e^0 \right) + \kappa\right) \,$ .

Then we can reduce to:

$f=2/3\left(1-e^{-3x}\right) + e^{-3x}\kappa \,$ .

Assume that kappa is 2 from the initial condition.

For a first-lodge linear ODE, with coefficients that may or may non vary with x:

$y'(10) + p (x) y (x) = r (x)$

Then,

$y=e^{-a(x)}\left(\int r(x) e^{a(x)} dx + \kappa\right)$

where $κ$ is the constant of integration, and

$a(x)=\int{p(x)dx}.$

This proof comes from Jean Bernoulli. Let

$y^\prime + py = r$

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

Then

$y^\prime = u^\prime v + u v^\prime$

Substituting into the differential equation,

$u^\prime v + u v^\prime + puv = r$

At present, the most important step: Since the differential equation is linear nosotros can carve up this into 2 contained equations and write

$u^\prime v + puv = 0$

$u v^\prime = r$

Since v is non aught, the top equation becomes

$u^\prime + pu = 0$

The solution of this is

$u = e^{ - \int p dx }$

Substituting into the second equation

$v = \int r e^{ \int p dx} + C$

Since y = uv, for capricious abiding C

$y =e^{ - \int p dx } \left( \int r e^{ \int p dx } + C \right)$

As an illustrative example, consider a starting time social club differential equation with constant coefficients:

$\frac{dy}{dx} + b y = 1.$

This equation is peculiarly relevant to get-go lodge systems such as RC circuits and mass-damper systems.

In this case, p(x) = b, r(x) = 1.

Hence its solution is

$y(x) = e^{-bx} \left( e^{bx}/b+ C \right) = 1/b + C e^{-bx} .$

Method of undetermined coefficients

The method of undetermined coefficients (MoUC), is useful in finding solution for $y p$ . Given the ODE $P (D) y = f (x)$ , discover another differential operator $A (D)$ such that $A (D) f (ten) = 0$ . 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 a homogeneous ODE $\big(A(D)P(D)\big)y = 0$ for which we find a solution basis $\{y_1,\ldots,y_n\}$ equally earlier. Then the original nonhomogeneous ODE is used to construct a arrangement of equations restricting the coefficients of the linear combinations to satisfy the ODE.

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

Case: Given $y'' - iv y' + v y = sin(m ten)$ , $P (D) = D 2 - 4 D + 5$ . The simplest annihilator of $sin(k x)$ is $A (D) = D 2 + k 2$ . The zeros of $A (z) P (z)$ are ${2 + i,ii - i, i k, - i thousand}$ , then the solution basis of $A (D) P (D)$ is ${y 1, y 2, y 3, y iv} = {e (2 + i) x, e (2 - i) x, due east i yard x, eastward - i thou 10}.$

Setting $y = c 1 y one + c 2 y 2 + c 3 y 3 + c 4 y 4$ we observe

$sin(thousand 10)$	$= P (D) y$
	$= P (D)(c ane y 1 + c 2 y + c three y iii + c 4 y 4)$
	$= c 1 P (D) y 1 + c 2 P (D) y two + c three P (D) y iii + c 4 P (D) y 4$
	$= 0 + 0 + c 3 ( - k 2 - 4 i k + v) y 3 + c four ( - k 2 + four i m + 5) y four$
	$= c 3 ( - one thousand 2 - 4 i g + 5)(cos(k x) + i sin(k ten)) + c 4 ( - g 2 + 4 i m + five)(cos(k ten) - i sin(g x))$

giving the system

i = (k 2 + 4 i grand - 5) c 3 + ( - k 2 + 4 i one thousand + v) c 4

0 = (k ii + iv i k - 5) c 3 + (grand 2 - four i k - 5) c 4

which has solutions

$c_3=\frac i{2(k^2+4ik-5)}$ , $c_4=\frac i{2(-k^2+4ik+5)}$

giving the solution set

Method of variation of parameters

Equally explained above, the full general solution to a non-homogeneous, linear differential equation $y''(x) + p (x) y'(x) + q (x) y (x) = chiliad (10)$ can be expressed equally the sum of the general solution $y h (ten)$ to the corresponding homogenous, linear differential equation $y''(x) + p (x) y'(10) + q (x) y (10) = 0$ and any ane solution $y p (x)$ to $y''(x) + p (x) y'(x) + q (x) y (ten) = g (x)$ .

Like the method of undetermined coefficients, described in a higher place, the method of variation of parameters is a method for finding one solution to $y''(ten) + p (x) y'(x) + q (x) y (10) = thousand (x)$ , having already found the general solution to $y''(x) + p (x) y'(10) + q (x) y (10) = 0$ . Unlike the method of undetermined coefficients, which fails except with certain specific forms of yard(x), the method of variation of parameters volition e'er piece of work; however, it is significantly more difficult to use.

For a second-social club equation, the method of variation of parameters makes utilise of the following fact:

Fact

Let p(10), q(x), and yard(x) exist functions, and let $y one (x)$ and $y 2 (x)$ exist solutions to the homogeneous, linear differential equation $y''(x) + p (x) y'(x) + q (x) y (x) = 0$ . Farther, let u(x) and five(x) be functions such that $u'(x) y 1 (10) + v'(x) y 2 (x) = 0$ and $u'(x) y 1'(x) + v'(x) y 2'(x) = g (x)$ for all ten, and define $y p (x) = u (ten) y 1 (10) + five (x) y ii (ten)$ . Then $y p (10)$ is a solution to the non-homogeneous, linear differential equation $y''(ten) + p (x) y'(x) + q (x) y (ten) = g (x)$ .

Proof

$y p (x) = u (x) y i (x) + 5 (x) y 2 (10)$

$y p'(x)$	$= u'(x) y 1 (x) + u (x) y i'(x) + five'(ten) y 2 (x) + v (10) y two'(ten)$
	$= 0 + u (x) y one'(x) + v (x) y 2'(x)$

$y p''(x)$	$= u'(x) y 1'(10) + u (10) y 1''(x) + v'(x) y 2'(x) + v (x) y 2''(x)$
	$= g (x) + u (ten) y i''(x) + v (x) y two''(x)$

$y p''(ten) + p (x) y' p (x) + q (x) y p (x) = grand (10) + u (x) y 1''(10) + v (x) y ii''(x) + p (x) u (x) y 1'(10) + p (x) five (x) y 2'(10) + q (x) u (x) y one (x) + q (x) five (ten) y 2 (ten)$

$= thou (x) + u (x)(y 1''(10) + p (x) y ane'(x) + q (10) y 1 (x)) + v (ten)(y two''(x) + p (x) y 2'(x) + q (x) y two (ten)) = thou (ten) + 0 + 0 = g (ten)$

Usage

To solve the 2d-order, non-homogeneous, linear differential equation $y''(x) + p (10) y'(x) + q (x) y (x) = g (10)$ using the method of variation of parameters, use the post-obit steps:

Detect the general solution to the corresponding homogeneous equation $y''(10) + p (x) y'(ten) + q (10) y (ten) = 0$ . Specifically, find two linearly contained solutions $y 1 (x)$ and $y two (x)$ .
Since $y 1 (x)$ and $y 2 (x)$ are linearly contained solutions, their Wronskian $y i (x) y 2'(ten) - y 1'(ten) y 2 (10)$ is nonzero, so we tin compute $- (1000 (10) y 2 (x)) / (y 1 (x) y 2'(x) - y one'(x) y 2 (x))$ and $(g (x) y 1 (x)) / (y 1 (x) y 2'(x) - y i'(ten) y 2 (ten))$ . If the erstwhile is equal to u'(x) and the latter to v'(ten), so u and five satisfy the ii constraints given above: that $u'(x) y 1 (ten) + v'(x) y 2 (x) = 0$ and that $u'(x) y 1'(x) + v'(x) y ii'(ten) = yard (x)$ . Nosotros tin can tell this after multiplying by the denominator and comparing coefficients.
Integrate $- (yard (x) y two (10)) / (y one (x) y 2'(10) - y one'(10) y 2 (x))$ and $(1000 (x) y 1 (x)) / (y 1 (x) y 2'(x) - y 1'(x) y 2 (10))$ to obtain u(x) and v(x), respectively. (Note that we only need one choice of u and v, so at that place is no need for constants of integration.)
Compute $y p (x) = u (x) y i (x) + v (x) y 2 (x)$ . The function $y p$ is 1 solution of $y''(ten) + p (x) y'(ten) + q (x) y (x) = g (10)$ .
The general solution is $c 1 y 1 (x) + c 2 y 2 (x) + y p (ten)$ , where $c 1$ and $c two$ are arbitrary constants.

College-lodge equations

The method of variation of parameters tin can also exist used with higher-order equations. For example, if $y 1 (x)$ , $y 2 (10)$ , and $y 3 (x)$ are linearly independent solutions to $y'''(ten) + p (x) y''(x) + q (x) y'(10) + r (x) y (x) = 0$ , then there exist functions u(x), v(ten), and westward(10) such that $u'(x) y 1 (x) + five'(x) y 2 (x) + w'(x) y 3 (x) = 0$ , $u'(x) y 1'(x) + v'(ten) y 2'(x) + w'(x) y 3'(x) = 0$ , and $u'(x) y 1''(x) + v'(x) y two''(x) + w'(10) y iii''(x) = g (10)$ . Having establish such functions (past solving algebraically for u'(x), v'(x), and westward'(ten), then integrating each), we have $y p (x) = u (x) y one (x) + v (10) y 2 (x) + due west (ten) y 3 (10)$ , one solution to the equation $y'''(10) + p (10) y''(x) + q (10) y'(x) + r (10) y (10) = g (10)$ .

Example

Solve the previous case, $y'' + y = sec x$ Call back $\sec x = \frac{1}{{\cos x}} = f$ . From technique learned from 3.1, LHS has root of $r = \pm i$ that yield $y c = C 1 cos x + C two sin x$ , (and so $y 1 = cos x$ , $y 2 = sin ten$ ) and its derivatives

$\left\{ {\begin{matrix} {\dot u = \frac{{ - y_2 f}}{W} = \frac{{ - \sin x}}{{\cos x}} = \tan x} \\ {\dot v = \frac{{y_1 f}}{W} = \frac{{\cos x}}{{\cos x}} = 1} \\ \end{matrix}} \right.$

where the Wronskian

$W\left( {y_1,y_2 :x} \right) = \left| {\begin{matrix} {\cos x} & {\sin x} \\ { - \sin x} & {\cos x} \\ \end{matrix}} \right| = 1$

were computed in order to seek solution to its derivatives.

Upon integration,

$\left\{ \begin{matrix} u = - \int {\tan x\,dx = - \ln \left| {\sec x} \right| + C} \\ v = \int {1\,dx = x + C} \\ \end{matrix} \right.$

Computing $y p$ and $y G$ :

$\begin{matrix} y_p = f = uy_1 + vy_2 = \cos x\ln \left| {\cos x} \right| + x\sin x \\ y_G = y_c + y_p = C_1 \cos x + C_2 \sin x + x\sin x + \cos x\ln \left( {\cos x} \right) \\ \end{matrix}$

Linear ODEs with variable coefficients

A linear ODE of order due north with variable coefficients has the full general grade

$p_{n}(x)y^{(n)}(x) + p_{n-1}(x) y^{(n-1)}(x) + \ldots + p_0(x) y(x) = r(x).$

Examples

A particular unproblematic example is the Cauchy-Euler equation often used in engineering

$x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \ldots + a_0 y(x) = 0.$

Theories of ODEs

Atypical solutions

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

Reduction to quadratures

The archaic endeavour in dealing with differential equations had in view a reduction to quadratures. As it had been the hope of eighteenth-century algebraists to notice a method for solving the general equation of the $due north$ th degree, so information technology was the promise of analysts to detect a general method for integrating whatsoever differential equation. Gauss (1799) showed, however, that the differential equation meets its limitations very soon unless circuitous numbers are introduced. Hence analysts began to substitute the study of functions, thus opening a new and fertile field. Cauchy was the showtime to capeesh the importance of this view. Thereafter the real question was to be, non whether a solution is possible by means of known functions or their integrals, merely 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 backdrop of this function.

Fuchsian theory

Ii memoirs by Fuchs (Crelle, 1866, 1868), inspired a novel approach, subsequently elaborated by Thomé and Frobenius. Collet was a prominent contributor first 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. Every bit the latter can be classified according to the backdrop of the fundamental curve which remains unchanged nether a rational transformation, so Clebsch proposed to allocate the transcendent functions divers by the differential equations according to the invariant backdrop of the corresponding surfaces f = 0 under rational one-to-one transformations.

Lie's theory

From 1870 Prevarication's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, past the introduction of what are at present called Prevarication groups, be referred to a common source; and that ordinary differential equations which admit the aforementioned minute transformations present comparable difficulties of integration. He likewise emphasized the subject field of transformations of contact (Berührungstransformationen).

Sturm-Liouville theory

Sturm-Liouville theory is a general method for resolution of second order linear equations with variable coefficients.

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Source: https://www.cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/o/Ordinary_differential_equation.htm

Given an Example of a Family of Differential

2007 Schools Wikipedia Selection. Related subjects: Mathematics

Definitions

Examples

Reduction of dimension

Types of ordinary differential equations

Homogeneous linear ODEs with constant coefficients