NationMaster - Encyclopedia: Examples of differential equations

where f(t) is some known function. We may solve this by separation of variables (moving the y terms to one side and the t terms to the other side), In mathematics, a function returns a unique output for a given input. ... In mathematics, separation of variables is any of several methods of solving ordinary and partial differential equations. ...

Integrating, we find In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

$ln y = -F(t) + C,$

where

$F(t) = int f(t),dt$

is the antiderivative of f(t) and C is a constant. Then, by exponentiation, we obtain In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

with A an arbitrary constant. (We can easily confirm that this is a solution by plugging it into the original differential equation.)

Some elaboration is needed since f(t) is not in fact a constant, indeed it might not even be integrable. Arguably, one must also assume something about the domains of the functions involved before the equation is fully defined. Are we talking complex functions, or just real, for example? The usual textbook approach is to discuss forming the equations well before considering how to solve them.

Non-separable first order linear ordinary differential equations

Some first order linear ODEs (ordinary differential equations) are not separable like in the above example. In order to solve non-separable first order linear ODEs one must use what is known as an integrating factor. This technique will be shown below. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, one solves certain ordinary differential equations by using an integrating factor. ...

Consider first order linear ODEs of the general form:

$frac{dy}{dx} + p(x)y = q(x)$

The method for solving this equation relies on a special "integrating factor", μ:

$mu = e^{int_{}^{} p(x), dx}$

Multiply both sides of the differential equation by μ to get:

$mu{frac{dy}{dx}} + mu{p(x)y} = mu{q(x)}$

Because of the special μ we picked, this simplifies to:

$mu{frac{dy}{dx}} + y{frac{d{mu}}{dx}} = mu{q(x)}$

Using the product rule we get: In mathematics, the product rule of calculus, which is also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

$frac{d}{dx}{(mu{y})} = mu{q(x)}$

Integrating both sides we get:

$mu{y} = left(intmu q(x), dxright) + C$

Finally, to solve for $y$ we divide both sides by $μ$ :

$y = frac{left(intmu q(x), dxright) + C}{mu}$

(Since μ is a function of x, we cannot simplify any more.)

A simple mathematical model

Suppose a mass is attached to a spring, which exerts an attractive force on the mass proportional to the extension/compression of the spring and ignore any other forces (gravity, friction etc). We shall write the extension of the spring at a time $t$ as $x (t)$ . Now, using Newton's second law we can write (using convenient units) The word proportionality may have one of a number of meanings: In mathematics, proportionality is a mathematical relation between two quantities. ... Gravity is a force of attraction that acts between bodies that have mass. ... Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. ... Newtons first and second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

$frac{d^2x}{dt^2} = - x$

If we look for solutions that have the form $C e k t$ , where $C$ is a constant, we discover the relationship $k 2 + 1 = 0$ , and thus $k$ must be one of the complex numbers $i$ or $- i$ . Thus, using Euler's theorem we can say that the solution must be of the form: In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (âˆ’1), which cannot be represented by any real number. ... This article is about the Eulers formula in complex analysis. ...

x (t) = A cos t + B sin t

To fix the unknown constants $A$ and $B$ , we need initial conditions, i.e. to specify the state of the system at a given time (usually taken to be $t = 0$ ).

For example, if we suppose at $t = 0$ the extension is a unit distance ( $x = 1$ ), and the particle is not moving ( $d x / d t = 0$ ). We have

x (0) = A cos0 + B sin0 = A = 1,

and so $A = 1$ .

x'(0) = - A sin0 + B cos0 = B = 0,

and so $B = 0$ .

Therefore $x (t) = cos t$ . This is an example of simple harmonic motion. Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ...

Improving our model

The above model of an oscillating mass on a spring is plausible but not really realistic. For a start, we have invented a perpetual motion machine which violates the second law of thermodynamics. Therefore, consider adding some friction for realism. Now, experimental scientists will tell us that friction will tend to decelerate the mass and have magnitude proportional to its velocity (i.e. $d x / d t$ ). Our new differential equation, expressing the balancing of the acceleration and the forces, is Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of temperature, pressure, and volume changes on physical systems at the macroscopic scale. ... Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. ... A scientist is a man who is an expert in at least one area of science and who uses the scientific method to do research. ...

$frac{d^2x}{dt^2} = - c frac{dx}{dt} - x$

where $c$ is our coefficient of friction, and $c > 0$ . Again looking for solutions of the form $A e k t$ , we find that

k 2 + c k + 1 = 0.

This is a quadratic equation which we can solve. If $c < 2$ we have complex roots $a pm i b$ , and the solution (with the above boundary conditions) will look like this: Graph of a quadratic function: y = x2 - x - 2 = (x+1)(x-2) The x-coordinates of the points where the graph crosses the x-axis, x = -1 and x = 2, are the roots of the quadratic equation: x2 - x - 2 = 0 In mathematics, a quadratic equation is a polynomial...

$x(t) = e^{at} left(cos bt - frac{a}{b} sin bt right)$

(We can show that $a < 0$ )

This is a damped oscillator, and the plot of displacement against time would look something like this:

which does resemble how we'd expect a vibrating spring to behave as friction removed the energy from the system. Wikipedia does not have an article with this exact name. ...

A simple exact equation

An exact differential equation is a first-order ordinary differential equation of implicit form In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

$I(x, y), dx + J(x, y), dy = 0, ,!$

such that

$frac{partial I}{partial y}(x, y) = frac{partial J}{partial x}(x, y).$

This equation has the solution

$int_{x_0}^{x} I(u, y), du + int_{y_0}^{y} J_partial (v), dv = 0$

where

$J_partial (y) = J(x, y) - frac{partial}{partial y} int_{x_0}^{x} I(u, y), du,$

u and v being dummy variables; x₀ and y₀ being initial-value constants. In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...

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 1584882972.

External link

Ordinary Differential Equations at EqWorld: The World of Mathematical Equations.

Category: Ordinary differential equations

Results from FactBites:

Ordinary differential equation - Wikipedia, the free encyclopedia (2832 words)
	In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable.
	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.
	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.

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