NationMaster - Encyclopedia: Linear differential equation

In mathematics, a linear differential equation is a differential equation of the form Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... An illustration of a differential equation. ...

Ly = f,

where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. Therefore a fairly general form of such an equation would be In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, LHS is informal shorthand for the left-hand side of an equation. ... In mathematics, a derivative is the rate of change of a quantity. ...

$a_n(x) D^n y(x) + a_{n-1}(x)D^{n-1} y(x) + cdots + a_1(x) D y(x) + a_0(x) y(x) =f(x)$

where D is the differential operator d/dx (i.e. Dy = y' , D²y = y",... ), and the a_i are given functions. Such an equation is said to have order n, the index of the highest derivative of f that is involved. (Assuming a possibly existing coefficient a_n of this derivative to be non zero, it is eliminated by dividing through it. In case it can become zero, different cases must be considered separately for the analysis of the equation.)

If y is assumed to be a function of only one variable, one speaks about an ordinary differential equation, else the derivatives and their coefficients must be understood as (contracted) vectors, matrices or tensors of higher rank, and we have a (linear) partial differential equation. In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

The case where f = 0 is called a homogeneous equation, and is particularly important to the solution of the general case (by a method traditionally called particular integral and complementary function). When the a_i are numbers, the equation is said to have constant coefficients. In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. ...

1 Homogeneous equations with constant coefficients
- 1.1 Examples
  - 1.1.1 Simple harmonic oscillator
  - 1.1.2 Damped harmonic oscillator
2 Nonhomogeneous equation with constant coefficients
- 2.1 Example
3 Equation with variable coefficients
- 3.1 Examples
4 First order equation
- 4.1 Examples
5 See also

Homogeneous equations with constant coefficients

To solve such an equation one makes a substitution

$y = e^{lambda x} !$ ,

to form the characteristic equation

${lambda^n +a_{n-1}lambda^{n-1}+cdots+a_1lambda+a_0 = 0}$

to obtain the solutions

$lambda=s_0, s_1, dots, s_{n-1}.$

When this polynomial has distinct roots, we have immediately n solutions to the differential equation in the form In mathematics, a polynomial P(X) is separable over a field K if its roots in an algebraic closure of K are distinct - that is P(X) has distinct linear factors in some large enough field extension. ...

${y_i(x)=e^{s_i x}.}$

It can be shown that these are linearly independent, by applying the Vandermonde determinant. Since homogenous linear DEs obey the superposition principle, their linear combinations, with n coefficients, should provide a complete solution. So it proves: it is known that the general solution to the homogeneous equation can be formed from a linear combination of the y_i, ie., In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ... In linear algebra, a Vandermonde matrix is a matrix with a geometric progression in each row, i. ... In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

${y_H(x)=A_0 y_0(x)+A_1 y_1+cdots+A_{n-1} y_{n-1}}$

Where the solutions are not distinct, it may be necessary to multiply them by some power of x to obtain linear independence; the general solution therefore involves the product of polynomials, of degrees bounded in terms of the multiplicities of the roots, and exponentials.

The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form $e z x$ , for possibly-complex values of $z$ . Thus to solve Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

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

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

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

so dividing by $e z x$ gives the nth-order polynomial

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

In short the terms

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

of the original differential equation are replaced by z^k. Solving the polynomial gives n values of z, $z_1, dots,z_n$ . Plugging those values into $e^{z_i x}$ gives a basis for the solution; any linear combination of these basis functions will satisfy the differential equation. A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. ... In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

This equation F(z) = 0, is the "characteristic" equation considered later by Monge and Cauchy. Gaspard Monge. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

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 2). 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 (possibly not real) zero(or root) of F(z) of multiplicity m and $kin{0,1,dots,m-1} ,$ then $y=x^ke^{zx} ,$ is a solution of the ODE. These functions make up a basis of the ODE's solutions. In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

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). In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...

A case that involves complex roots can be solved with the aid of Euler's formula. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

Examples

Given $y''-4y'+5y=0 ,$ . The characteristic equation is $z^2-4z+5=0 ,$ which has zeroes 2+i and 2−i. Thus the solution basis ${y 1, y 2}$ is ${e^{(2+i)x},e^{(2-i)x}} ,$ . Now y is a solution iff $y=c_1y_1+c_2y_2 ,$ for $c_1,c_2inmathbb C$ . IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

Because the coefficients are real,

we are likely not interested in the complex solutions
our basis 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 us a real basis in ${u 1, u 2}$ .

Simple harmonic oscillator

The second order differential equation

D 2 y = - k 2 y,

which represents a simple harmonic oscillator, can be restated as In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ...

(D 2 + k 2) y = 0.

The expression in parenthesis can be factored out, yielding

(D + i k)(D - i k) y = 0,

which has a pair of linearly independent solutions, one for

(D - i k) y = 0

and another for

(D + i k) y = 0.

The solutions are, respectively,

y 0 = A 0 e i k x

and

y 1 = A 1 e - i k x .

These solutions provide a basis for the two-dimensional "solution space" of the second order differential equation: meaning that linear combinations of these solutions will also be solutions. In particular, the following solutions can be constructed In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

$y_{0'} = {A_0 e^{i k x} + A_0 e^{-i k x} over 2} = A_0 cos (k x)$

and

$y_{1'} = {A_1 e^{i k x} - A_1 e^{-i k x} over 2 i} = A_1 sin (k x).$

These last two trigonometric solutions are linearly independent, so they can serve as another basis for the solution space, yielding the following general solution:

y H = A 0 cos(k x) + A 1 sin(k x).

Damped harmonic oscillator

Given the equation for the damped harmonic oscillator: In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ...

$left(D^2 + {b over m} D + omega_0^2right) y = 0,$

the expression in parentheses can be factored out: first obtain the characteristic equation by replacing D with λ. This equation must be satisfied for all y, thus:

$lambda^2 + {b over m} lambda + omega_0^2 = 0.$

Solve using the quadratic formula: In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

$lambda = {-b/m pm sqrt{b^2 / m^2 - 4 omega_0^2} over 2}.$

Use these data to factor out the original differential equation:

$left(D + {b over 2 m} - sqrt{{b^2 over 4 m^2} - omega_0^2} right) left(D + {b over 2m} + sqrt{{b^2 over 4 m^2} - omega_0^2}right) y = 0.$

This implies a pair of solutions, one corresponding to

$left(D + {b over 2 m} - sqrt{{b^2 over 4 m^2} - omega_0^2} right) y = 0$

and another to

$left(D + {b over 2m} + sqrt{{b^2 over 4 m^2} - omega_0^2}right) y = 0$

The solutions are, respectively,

$y_0 = A_0 e^{-omega x + sqrt{omega^2 - omega_0^2} x} = A_0 e^{-omega x} e^{sqrt{omega^2 - omega_0^2} x}$

and

$y_1 = A_1 e^{-omega x - sqrt{omega^2 - omega_0^2} x} = A_1 e^{-omega x} e^{-sqrt{omega^2 - omega_0^2} x}$

where ω = b / 2m. From this linearly independent pair of solutions can be constructed another linearly independent pair which thus serve as a basis for the two-dimensional solution space:

$y_H (A_0, A_1) (x) = left(A_0 sinh sqrt{omega^2 - omega_0^2} x + A_1 cosh sqrt{omega^2 - omega_0^2} xright) e^{-omega x}.$

However, if |ω| < |ω₀| then it is preferable to get rid of the consequential imaginaries, expressing the general solution as

$y_H (A_0, A_1) (x) = left(A_0 sin sqrt{omega_0^2 - omega^2} x + A_1 cos sqrt{omega_0^2 - omega^2} xright) e^{-omega x}.$

This latter solution corresponds to the underdamped case, whereas the former one corresponds to the overdamped case: the solutions for the underdamped case oscillate whereas the solutions for the overdamped case do not. Oscillation is the variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ...

Nonhomogeneous equation with constant coefficients

To obtain the solution to the non-homogeneous equation (sometimes called inhomogeneous equation), find a particular solution y_P(x) by either the method of undetermined coefficients or the method of variation of parameters; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular solution. In mathematics, the method of undetermined coefficients is an approach to solving certain ordinary differential equations and recurrence relations. ... In mathematics, variation of parameters is a technique used in solving certain second order linear inhomogeneous ordinary differential equations. ...

Suppose we face

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

For later convenience, define the characteristic polynomial

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

We find the solution basis ${y_1,y_2,ldots,y_n}$ 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:

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

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

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

But $P (D) y j = 0$ , 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 system in the $u' j$ . This much can always be solved; in fact, combining Cramer's rule with the Wronskian, Cramers rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. ... In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations. ...

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

The rest is a matter of integrating $u' j .$

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

Example

Suppose $y'' - 4 y' + 5 y = s i n (k x)$ . We take the solution basis found above ${e (2 + i) x, e (2 - i) x}$ .

$W,$	$= begin{vmatrix}e^{(2+i)x}&e^{(2-i)x} (2+i)e^{(2+i)x}&(2-i)e^{(2-i)x} end{vmatrix}$
	$=e^{4x}begin{vmatrix}1&1 2+i&2-iend{vmatrix}$
	$=-2ie^{4x},$

$u'_1,$	$=frac{1}{W}begin{vmatrix}0&e^{(2-i)x} sin(kx)&(2-i)e^{(2-i)x}end{vmatrix}$
	$=-frac{i}2sin(kx)e^{(-2-i)x}$

$u'_2,$	$=frac{1}{W}begin{vmatrix}e^{(2+i)x}&0 (2+i)e^{(2+i)x}&sin(kx)end{vmatrix}$
	$=frac{i}{2}sin(kx)e^{(-2+i)x}.$

Using the list of integrals of exponential functions The following is a list of integrals (antiderivative functions) of exponential functions. ...

$u_1,$	$=-frac{i}{2}intsin(kx)e^{(-2-i)x},dx$
	$=frac{ie^{(-2-i)x}}{2(3+4i+k^2)}left((2+i)sin(kx)+kcos(kx)right)$

$u_2,$	$=frac i2intsin(kx)e^{(-2+i)x},dx$
	$=frac{ie^{(i-2)x}}{2(3-4i+k^2)}left((i-2)sin(kx)-kcos(kx)right).$

And so

$y_p,$	$=frac{i}{2(3+4i+k^2)}left((2+i)sin(kx)+kcos(kx)right) +frac{i}{2(3-4i+k^2)}left((i-2)sin(kx)-kcos(kx)right)$
	$=frac{(5-k^2)sin(kx)+4kcos(kx)}{(3+k^2)^2+16}.$

(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 $c 1 y 1 + c 2 y 2$ is the transient. In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ...

Equation with variable coefficients

A linear ODE of order n with variable coefficients has the general form

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

Examples

A particular simple example is the Cauchy-Euler equation often used in engineering In mathematics, a Cauchy-Euler equation (also Euler-Cauchy equation) is a second-order ordinary differential equation of the form These differential equations have one relatively simple solution xÎ±. Observe Since xÎ± is zero only when x is zero for positive Î± (which corresponds to a trivial solution) and never zero...

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

First order equation

Example

$-3y''+4y'=5 ,$

with the initial condition

$fleft(0right)=2 ,$ .

Using the general solution method:

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

The integration is done from 0 to x, giving:

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

Then we can reduce to:

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

Assume that kappa is 2 from the initial condition.

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

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

Then,

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

where $κ$ is the constant of integration, and

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

This proof comes from Jean Bernoulli. Let Categories: People stubs | 1667 births | 1748 deaths | Swiss mathematicians ...

$y^prime + py = r$

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,

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

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

$u^prime v + puv = 0$

$u v^prime = r$

Since v is not zero, 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 arbitrary constant C

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

Examples

Consider a first order differential equation with constant coefficients: In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. ...

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

This equation is particularly relevant to first order systems such as RC circuits and mass-damper systems. A resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue electronic filters. ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ...

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

Hence its solution is

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

Contents

Homogeneous equations with constant coefficients GA_googleFillSlot("encyclopedia_square");

Examples

Simple harmonic oscillator

Damped harmonic oscillator

Nonhomogeneous equation with constant coefficients

Example

Equation with variable coefficients

Examples

First order equation

Examples

See also

Homogeneous equations with constant coefficients