ODE redirects here. For the real-time physics engine, see open dynamics engine.

In mathematics, and particularly in analysis, 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. See differential calculus and integral calculus for basic calculus background.

Many scientific theories can be expressed clearly and concisely in terms of ordinary differential equations. For instance, the law for radioactive decay of a single isotope of an element, states that its rate of loss of mass is proportional to its mass. If t represents time and u(t) represents the mass of the isotope at time t, then the law for decay states that

$\backslash frac\{du\}\{dt\}\; =\; -\backslash alpha\; u\backslash ,$

where $\backslash alpha$ is a constant that depends upon the particular isotope.

Another example of an ordinary differential equation is Newton's second law of motion of a single particle, which states that F = ma, where F is an applied force, m is the mass of the particle, and a is the acceleration of the particle due to the force. If motion is constrained to a straight line, the coordinate t measures the time elapsed and the unknown function x(t) specifies the position of the particle along the line, then the velocity of the particle v is given by the first derivative of x with respect to t:

$v=\backslash frac\{dx\}\{dt\}\backslash ,$.

Similarly, the acceleration of the particle a is given by the second derivative of x:

$a\; =\; \backslash frac\{dv\}\{dt\}\; =\; \backslash frac\{d^2\; x\}\{dt^2\}\backslash ,$.

Thus Newton's second law implies the differential equation

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

In general, the force depends upon the position of the particle, and thus the unknown variable x appears on both sides of the differential equation, as is indicated in the notation f(x).

Important theorems in the field of ODEs include broad existence and uniqueness theorems and for ODEs in the plane, the Poincaré-Bendixson theorem.

Definition

---

Let y represent an unknown function of x, and let

$y\text{'},\; y\text{'}\text{'},\backslash \; \backslash dots,\backslash \; y^\{(n)\}$

denote the derivatives

$\backslash frac\{dy\}\{dx\},\backslash \; \backslash frac\{d^\{2\}y\}\{dx^2\},\backslash \; \backslash dots,\backslash \; \backslash frac\{d^\{n\}y\}\{dx^\{n\}\}.$

An ordinary differential equation (ODE) is an equation involving

$x,\backslash \; y,\backslash \; y\text{'},\backslash \; y\text{'}\text{'},\backslash \; \backslash dots\; .$

The order of a differential equation is the order n of the highest derivative that appears. If the highest derivative appears only in integer powers, then the degree of the equation is the highest power of the highest derivative.

A solution of an ODE is a function y(x) whose derivatives satisfy the equation. Such a function is not guaranteed to exist and, if it does exist, is usually not unique. A general solution of an nth-order equation is a solution containing $n$ arbitrary variables, corresponding to n constants of integration. A particular solution is derived from the general solution by setting the constants to particular values. A singular solution is a solution that can't be derived from the general solution.

When a differential equation of order n has the form

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

it is called an implicit differential equation whereas the form

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

is called an explicit differential equation.

A differential equation not depending on x is called autonomous, and one with no terms depending only on x is called homogeneous.

General application

---

An important special case is when the equations do not involve $x$. These differential equations may be represented as vector fields. This type of differential equation has the property that space can be divided into equivalence classes based on whether two points lie on the same solution curve. Since the laws of physics are believed not to change with time, the physical world is governed by such differential equations. (See also symplectic topology for abstract discussion.)

In the case where the equations are linear, the original equation can be solved by breaking it down into smaller equations, solving those, and then adding the results back together. Unfortunately, many of the interesting differential equations are non-linear, which means that they cannot be broken down in this way. There are also a number of techniques for solving differential equations using a computer (see numerical ordinary differential equations).

Ordinary differential equations are to be distinguished from partial differential equations where $y$ is a function of several variables, and the differential equation involves partial derivatives.

Existence and nature of solutions

---

The problem of solving a differential equation is to find the function $y$ whose derivatives satisfy the equation. For example, the differential equation

$y\text{'}\text{'}\; +\; y\; =\; 0\; \backslash ,$

has the general solution

$y\; =\; A\; \backslash cos\{x\}\; +\; B\; \backslash sin\{x\}\; \backslash ,$,

where A, B are constants determined from boundary conditions.

In general, an n-th order equation allows both $x$ and $y$ to be fixed, as well as all the $n-1$ lower order derivatives of $y$; the remaining equation can be solved (at least conceptionally) for $y^\{(n)\}$. If the equation has finite degree $d$, then we now have a polynomial equation in $y^\{(n)\}$ with at most $d$ roots. Therefore there can be as many as $d$ possible values for $y^\{(n)\}$ at any given point and for any possible values of the lower order derivatives, though there may be ranges of these points and values where there are fewer solutions (or none at all). A Lipschitz condition must also be satisfied for a solution to exist.

Thus, in the previous example, a second-order, first-degree equation, any point on the plane and any slope through that point can be selected and yield a unique solution (since the single root of $y\text{'}\text{'}$ exists for any value of $y$). Note in particular that there are an infinity of solutions through any given point; this is a general characteristic of equations of order higher than one.

Consider now

$(y\text{'})^2\; +\; xy\text{'}\; -\; y\; =\; 0\; \backslash ,$

with general solution

$y\; =\; Ax\; +\; A^2\; \backslash ,$

This is a first-order, second-degree equation, thus any point can have at most two solutions passing through it, corresponding to the two roots of $y\text{'}$ in the quadratic equation that would result after fixing $x$ and $y$. Studying the quadratic equation's discriminant ($x^2\; +\; 4y$) leads to the conclusion that only a single solution exists along the parabola $y\; =\; -\; \backslash frac\{1\}\{4\}\; x^2$ (where the discriminant is zero) and that no solution exists below this parabola (where both roots are complex).

The parabola in this problem is an example of a cusp locus; a curve along which two or more roots of the differential equation are identical. Along such a locus it is possible to move from one general solution to another while still obeying the differential equation; thus the presence of cusp loci introduce the possibility of singular solutions. In this example, the parabola $y\; =\; -\; \backslash frac\{1\}\{4\}\; x^2$ is such a singular solution; it satisfies the original differential equation, and a full set of solutions must include such possibilities as the hybrid solution:

where the cusp locus has been used to connect two particular solutions; note that the first derivative (the only derivative to appear in the differential equation) is continuous at the transitions.

(Johnson, Chapter 5)

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.

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 $e^\{z\; x\}$, for possibly-complex values of $z$. Thus

$\backslash frac\; \{d^\{n\}y\}\; \{dx^\{n\}\}\; +\; A\_\{1\}\backslash frac\; \{d^\{n-1\}y\}\; \{dx^\{n-1\}\}\; +\; \backslash cdots\; +\; A\_\{n\}y\; =\; 0$

has the form

$z^n\; e^\{zx\}\; +\; A\_1\; z^\{n-1\}\; e^\{zx\}\; +\; \backslash cdots\; +\; A\_n\; e^\{zx\}\; =\; 0$

so dividing by $e^\{zx\}$ gives the nth-order polynomial

$F(z)\; =\; z^\{n\}\; +\; A\_\{1\}z^\{n-1\}\; +\; \backslash cdots\; +\; A\_n\; =\; 0$

In short the terms

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

of the original differential equation are replaced by z^k. Solving the polynomial gives n values of z, $z\_1,\; \backslash dots,z\_n$. 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.

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

If z is a (possibly not real) zero of F(z) of multiplicity m and $k\backslash in\backslash \{0,1,\backslash dots,m-1\backslash \}\; \backslash ,$ then $y=x^ke^\{zx\}\; \backslash ,$ 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 $y$-4y'+5y=0 \,. The characteristic equation is $z^2-4z+5=0\; \backslash ,$ which has zeroes 2+i and 2−i. Thus the solution basis $\backslash \{y\_1,y\_2\backslash \}$ is $\backslash \{e^\{(2+i)x\},e^\{(2-i)x\}\backslash \}\; \backslash ,$. Now y'' is a solution iff $y=c\_1y\_1+c\_2y\_2\; \backslash ,$ for $c\_1,c\_2\backslash in\backslash mathbb\; C$.

Because the coefficients are real,

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

The linear combinations

$u\_1=\backslash mbox\{Re\}(y\_1)=\backslash frac\{y\_1+y\_2\}\{2\}=e^\{2x\}\backslash cos(x)\; \backslash ,$ and

$u\_2=\backslash mbox\{Im\}(y\_1)=\backslash frac\{y\_1-y\_2\}\{2i\}=e^\{2x\}\backslash sin(x)\; \backslash ,$

will give us a real basis in $\backslash \{u\_1,u\_2\backslash \}$.

Linear ODEs with constant coefficients

Suppose instead we face

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

For later convenience, define the characteristic polynomial

$P(v)=v^n+A\_1v^\{n-1\}+\backslash cdots+A\_n.$

We find the solution basis $\backslash \{y\_1,y\_2,\backslash ldots,y\_n\backslash \}$ 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:

$y\_p=u\_1y\_1+u\_2y\_2+\backslash cdots+u\_ny\_n.$

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

$f=P(D)y\_p=P(D)(u\_1y\_1)+P(D)(u\_2y\_2)+\backslash cdots+P(D)(u\_ny\_n).$

With the constraints

$0=u\text{'}\_1y\_1+u\text{'}\_2y\_2+\backslash cdots+u\text{'}\_ny\_n$

$0=u\text{'}\_1y\text{'}\_1+u\text{'}\_2y\text{'}\_2+\backslash cdots+u\text{'}\_ny\text{'}\_n$

…

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

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

$f=u\_1P(D)y\_1+u\_2P(D)y\_2+\backslash cdots+u\_nP(D)y\_n+u\text{'}\_1y^\{(n-1)\}\_1+u\text{'}\_2y^\{(n-1)\}\_2+\backslash cdots+u\text{'}\_ny^\{(n-1)\}\_n.$

But $P(D)y\_j=0$, therefore

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

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

$u\text{'}\_j=(-1)^\{n+j\}\backslash frac\{W(y\_1,\backslash ldots,y\_\{j-1\},y\_\{j+1\}\backslash ldots,y\_n)\_\{0\; \backslash choose\; f\}\}\{W(y\_1,y\_2,\backslash ldots,y\_n)\}.$

The rest is a matter of integrating $u\text{'}\_j.$

The particular solution is not unique; $y\_p+c\_1y\_1+\backslash cdots+c\_ny\_n$ also satisfies the ODE for any set of constants c_j.

See also variation of parameters.

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

{| $W\backslash ,$ $=-2ie^\{4x\}\backslash ,$

{| $u\text{'}\_1\backslash ,$ $=-\backslash frac\{i\}2\backslash sin(kx)e^\{(-2-i)x\}$

{| $u\text{'}\_2\backslash ,$ $=\backslash frac\{i\}\{2\}\backslash sin(kx)e^\{(-2+i)x\}.$

Using the list of integrals of exponential functions

{| $u\_1\backslash ,$ $=-\backslash frac\{i\}\{2\}\backslash int\backslash sin(kx)e^\{(-2-i)x\}\backslash ,dx$ $=\backslash frac\{ie^\{(-2-i)x\}\}\{2(3+4i+k^2)\}\backslash left((2+i)\backslash sin(kx)+k\backslash cos(kx)\backslash right)$

{| $u\_2\backslash ,$ $=\backslash frac\; i2\backslash int\backslash sin(kx)e^\{(-2+i)x\}\backslash ,dx$ $=\backslash frac\{ie^\{(i-2)x\}\}\{2(3-4i+k^2)\}\backslash left((i-2)\backslash sin(kx)-k\backslash cos(kx)\backslash right).$

And so

{| $y\_p\backslash ,$ $=\backslash frac\{i\}\{2(3+4i+k^2)\}\backslash left((2+i)\backslash sin(kx)+k\backslash cos(kx)\backslash right)$

+\frac{i}{2(3-4i+k^2)}\left((i-2)\sin(kx)-k\cos(kx)\right) $=\backslash frac\{(5-k^2)\backslash sin(kx)+4k\backslash cos(kx)\}\{(3+k^2)^2+16\}.$ (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 $c\_1y\_1+c\_2y\_2$ is the transient.

Linear ODEs with variable coefficient

Method of undetermined coefficients

The method of undetermined coefficients (MoUC), is useful in finding solution for $y\_p$. Given the ODE $P(D)y\; =\; f(x)$, 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 $\backslash big(A(D)P(D)\backslash big)y\; =\; 0$ for which we find a solution basis $\backslash \{y\_1,\backslash ldots,y\_n\backslash \}$ 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 $y\text{'}\text{'}-4y\text{'}+5y=\backslash sin(kx)$, $P(D)=D^2-4D+5$. The simplest annihilator of $\backslash sin(kx)$ is $A(D)=D^2+k^2$. The zeros of $A(z)P(z)$ are $\backslash \{2+i,2-i,ik,-ik\backslash \}$, so the solution basis of $A(D)P(D)$ is $\backslash \{y\_1,y\_2,y\_3,y\_4\backslash \}=\backslash \{e^\{(2+i)x\},e^\{(2-i)x\},e^\{ikx\},e^\{-ikx\}\backslash \}.$

Setting $y=c\_1y\_1+c\_2y\_2+c\_3y\_3+c\_4y\_4$ we find

{| $\backslash sin(kx)$ $=P(D)y$ $=P(D)(c\_1y\_1+c\_2y+c\_3y\_3+c\_4y\_4)$ $=c\_1P(D)y\_1+c\_2P(D)y\_2+c\_3P(D)y\_3+c\_4P(D)y\_4$ $=0+0+c\_3(-k^2-4ik+5)y\_3+c\_4(-k^2+4ik+5)y\_4$ $=c\_3(-k^2-4ik+5)(\backslash cos(kx)+i\backslash sin(kx))$

+c_4(-k^2+4ik+5)(\cos(kx)-i\sin(kx)) giving the system

$i=(k^2+4ik-5)c\_3+(-k^2+4ik+5)c\_4$

$0=(k^2+4ik-5)c\_3+(k^2-4ik-5)c\_4$

which has solutions

$c\_3=\backslash frac\; i\{2(k^2+4ik-5)\}$, $c\_4=\backslash frac\; i\{2(-k^2+4ik+5)\}$

giving the solution set

{| $y\backslash ,$ $=c\_1y\_1+c\_2y\_2+\backslash frac\; i\{2(k^2+4ik-5)\}y\_3+\backslash frac\; i\{2(-k^2+4ik+5)\}y\_4$ $=c\_1y\_1+c\_2y\_2+\backslash frac\{4k\backslash cos(kx)-(k^2-5)\backslash sin(kx)\}$

{(k^2+4ik-5)(k^2-4ik-5)} $=c\_1y\_1+c\_2y\_2\; +\backslash frac\{4k\backslash cos(kx)+(5-k^2)\backslash sin(kx)\}\{k^4+6k^2+25\}.$

Method of variation of parameters

.

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 $y\text{'}\text{'}(x)\; +\; p(x)\; y\text{'}(x)\; +\; q(x)\; y(x)\; =\; g(x)$.

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:

= Fact

= Let p(x), q(x), and g(x) be functions, and let $y\_1(x)$ and $y\_2(x)$ 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 $u\text{'}(x)\; y\_1(x)\; +\; v\text{'}(x)\; y\_2(x)\; =\; 0$ and $u\text{'}(x)\; y\_1\text{'}(x)\; +\; v\text{'}(x)\; y\_2\text{'}(x)\; =\; g(x)$ for all x, and define $y\_p(x)\; =\; u(x)\; y\_1(x)\; +\; v(x)\; y\_2(x)$. 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).

= Proof

= $y\_p(x)\; =\; u(x)\; y\_1(x)\; +\; v(x)\; y\_2(x)$

$y\_p\text{'}(x)$	$=\; u\text{'}(x)\; y\_1(x)\; +\; u(x)\; y\_1\text{'}(x)\; +\; v\text{'}(x)\; y\_2(x)\; +\; v(x)\; y\_2\text{'}(x)$
	$=\; 0\; +\; u(x)\; y\_1\text{'}(x)\; +\; v(x)\; y\_2\text{'}(x)$

$y\_p\text{'}\text{'}(x)$	$=\; u\text{'}(x)\; y\_1\text{'}(x)\; +\; u(x)\; y\_1$(x) + v'(x) y_2'(x) + v(x) y_2(x)
	$=\; g(x)\; +\; u(x)\; y\_1$(x) + v(x) y_2(x)

$y\_p$(x) + p(x) y'_p(x) + q(x) y_p(x) = g(x) + u(x) y_1(x) + v(x) y_2''(x) + p(x) u(x) y_1'(x) + p(x) v(x) y_2'(x) + q(x) u(x) y_1(x) + q(x) v(x) y_2(x)

$=\; g(x)\; +\; u(x)\; (y\_1$(x) + p(x) y_1'(x) + q(x) y_1(x)) + v(x) (y_2(x) + p(x) y_2'(x) + q(x) y_2(x)) = g(x) + 0 + 0 = g(x)

= Usage

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

1. Find the general solution to the corresponding homogeneous equation $y\text{'}\text{'}(x)\; +\; p(x)\; y\text{'}(x)\; +\; q(x)\; y(x)\; =\; 0$. Specifically, find two linearly independent solutions $y\_1(x)$ and $y\_2(x)$.
2. Since $y\_1(x)$ and $y\_2(x)$ are linearly independent solutions, their Wronskian $y\_1(x)\; y\_2\text{'}(x)\; -\; y\_1\text{'}(x)\; y\_2(x)$ is nonzero, so we can compute $-(g(x)\; y\_2(x))/(\{y\_1(x)\; y\_2\text{'}(x)\; -\; y\_1\text{'}(x)\; y\_2(x)\})$ and $(\{g(x)\; y\_1(x)\})/(\{y\_1(x)\; y\_2\text{'}(x)\; -\; y\_1\text{'}(x)\; y\_2(x)\})$. 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 $u\text{'}(x)\; y\_1(x)\; +\; v\text{'}(x)\; y\_2(x)\; =\; 0$ and that $u\text{'}(x)\; y\_1\text{'}(x)\; +\; v\text{'}(x)\; y\_2\text{'}(x)\; =\; g(x)$. We can tell this after multiplying by the denominator and comparing coefficients.
3. Integrate $-(g(x)\; y\_2(x))/(\{y\_1(x)\; y\_2\text{'}(x)\; -\; y\_1\text{'}(x)\; y\_2(x)\})$ and $(\{g(x)\; y\_1(x)\})/(\{y\_1(x)\; y\_2\text{'}(x)\; -\; y\_1\text{'}(x)\; y\_2(x)\})$ 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 $y\_p(x)\; =\; u(x)\; y\_1(x)\; +\; v(x)\; y\_2(x)$. The function $y\_p$ is one solution of $y\text{'}\text{'}(x)\; +\; p(x)\; y\text{'}(x)\; +\; q(x)\; y(x)\; =\; g(x)$.
5. The general solution is $c\_1\; y\_1(x)\; +\; c\_2\; y\_2(x)\; +\; y\_p(x)$, where $c\_1$ and $c\_2$ are arbitrary constants.

= Higher-order equations

= The method of variation of parameters can also be used with higher-order equations. For example, if $y\_1(x)$, $y\_2(x)$, and $y\_3(x)$ 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 $u\text{'}(x)\; y\_1(x)\; +\; v\text{'}(x)\; y\_2(x)\; +\; w\text{'}(x)\; y\_3(x)\; =\; 0$, $u\text{'}(x)\; y\_1\text{'}(x)\; +\; v\text{'}(x)\; y\_2\text{'}(x)\; +\; w\text{'}(x)\; y\_3\text{'}(x)\; =\; 0$, and $u\text{'}(x)\; y\_1$(x) + v'(x) y_2(x) + w'(x) y_3(x) = g(x). Having found such functions (by solving algebraically for u'(x), v'(x), and w'(x), then integrating each), we have $y\_p(x)\; =\; u(x)\; y\_1(x)\; +\; v(x)\; y\_2(x)\; +\; w(x)\; y\_3(x)$, one solution to the equation $y$(x) + p(x) y''(x) + q(x) y'(x) + r(x) y(x) = g(x).

= Example

= Solve the previous example, $y\text{'}\text{'}\; +\; y\; =\; \backslash sec\; x$ Recall $\backslash sec\; x\; =\; \backslash frac\{1\}\; =\; f$. From technique learned from 3.1, LHS has root of $r\; =\; \backslash pm\; i$ that yield $y\_c\; =\; C\_1\; \backslash cos\; x\; +\; C\_2\; \backslash sin\; x$, (so $y\_1\; =\; \backslash cos\; x$, $y\_2\; =\; \backslash sin\; x$ ) and its derivatives

$\backslash left\backslash \{\; \{\backslash begin\{matrix\}$

{\dot u = \frac{W} = \frac = \tan x} \\ {\dot v = \frac{W} = \frac = 1} \\ \end{matrix}} \right. where the Wronskian

$W\backslash left(\; \{y\_1,y\_2\; :x\}\; \backslash right)\; =\; \backslash left|\; \{\backslash begin\{matrix\}$

{\cos x} & {\sin x} \\ { - \sin x} & {\cos x} \\ \end{matrix}} \right| = 1 were computed in order to seek solution to its derivatives.

Upon integration,

$\backslash left\backslash \{\; \backslash begin\{matrix\}$

u = - \int {\tan xdx = - \ln \left| {\sec x} \right| + C} \\ v = \int {1dx = x + C} \\ \end{matrix} \right. Computing $y\_p$ and $y\_G$:

$\backslash begin\{matrix\}$

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}

General solution method for first-order linear ODEs

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

$x\text{'}(t)\; +\; p(t)\; x(t)\; =\; r(t)$

Then,

$x=e^\{-a(t)\}\backslash left(\backslash int\; r(t)\; e^\{a(t)\}\; dt\; +\; \backslash kappa\backslash right)$

where $\backslash kappa$ is the constant of integration, and

$a(t)=\backslash int\{p(s)ds\}.$

Proof

This proof comes from Jean Bernoulli. Let

$x^\backslash prime\; +\; px\; =\; r$

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

Then

$x^\backslash prime\; =\; u^\backslash prime\; v\; +\; u\; v^\backslash prime$

Substituting into the differential equation,

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

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

$u^\backslash prime\; v\; +\; puv\; =\; 0$

$u\; v^\backslash prime\; =\; r$

Since v is not zero, the top equation becomes

$u^\backslash prime\; +\; pu\; =\; 0$

The solution of this is

$u\; =\; e^\{\; -\; \backslash int\; p\; dt\; \}$

Substituting into the second equation

$v\; =\; \backslash int\; r\; e^\{\; \backslash int\; p\; dt\; \}\; +\; C$

Since x = uv, for arbitrary constant C

$x\; =e^\{\; -\; \backslash int\; p\; dt\; \}\; \backslash left(\; \backslash int\; r\; e^\{\; \backslash int\; p\; dt\; \}\; +\; C\; \backslash right)$

First order differential equation with constant coefficients

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

$a\backslash frac\{dx\}\{dt\}\; +\; bx\; =\; Af(t).$

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

After nondimensionalization, the equation becomes

$\backslash frac\{d\; \backslash chi\}\{d\; \backslash tau\}\; +\; \backslash chi\; =\; F(\backslash tau).$

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

Hence its solution by inspection is

$\backslash chi\; (\backslash tau)\; =\; e^\{-\backslash tau\}\; \backslash left(\; \backslash int\; F(\backslash tau)e^\{\backslash tau\}\; \backslash ,\; d\backslash tau\; +\; C\; \backslash right).$

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.

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 $n$th 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.

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.

Lie's theory

---

From 1870 Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact (Berührungstransformationen).

See also

---

• Examples of differential equations
• Differential equations of mathematical physics
• Differential equations from outside physics
• Difference equation
• Laplace transform applied to differential equations
• Boundary value problem
• List of dynamical systems and differential equations topics

Bibliography

---

• A. D. Polyanin and V. F. Zaitsev, ''Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition)", Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2
• A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations'', Taylor & Francis, London, 2002. ISBN 0-415-27267-X
• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.
• Hartman, Philip, Ordinary Differential Equations, 2nd Ed., Society for Industrial & Applied Math, 2002. ISBN 0898715105.
• W. Johnson, A Treatise on Ordinary and Partial Differential Equations, John Wiley and Sons, 1913, in University of Michigan Historical Math Collection

External links

---

• EqWorld: The World of Mathematical Equations, containing a list of ordinary differential equations with their solutions.

Differential calculus | Ordinary differential equations

Equació diferencial ordinària | Gewöhnliche Differentialgleichung | Equazione differenziale ordinaria | 常微分方程式 | Równanie różniczkowe zwyczajne | Equação diferencial ordinária | Ecuaţie diferenţială ordinară | Обыкновенные дифференциальные уравнения | Ordinär differentialekvation | 常微分方程