Examples of differential equations

Differential equations arise in many problems in physics, engineering, etc. The following examples show how to solve differential equations in a few simple cases when an exact solution exists.

eparable first order linear ordinary differential equations

A separable linear ordinary differential equation of the first order has the general form:

:frac{dy}{dt} + f(t) y = 0

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),

:frac{dy}{y} = -f(t), dt

Antidifferentiating, we find

:ln |y| = left(-int f(t),dt ight) + C,

where "C" is a constant. Then, by exponentiation, we obtain

:y = pm e^{left(-int f(t),dt ight) + C} = pm e^{-int f(t),dt} cdot e^{C} = A e^{-int f(t),dt}

with "A" another arbitrary constant. It is easy to confirm that this is a solution by plugging it into the original differential equation:

:frac{dy}{dt} + f(t) y = A e^{-int f(t),dt} cdot -f(t) + f(t) cdot A e^{-int f(t),dt} = 0

Some elaboration is needed because f(t) is not in necessarily 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 about 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". 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}

We choose this integrating factor because it has the special property that its derivative is itself times the function we are integrating, that is:

:frac{d{mu{dx} = e^{int_{}^{} p(x), dx} cdot p(x) = mu p(x)

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

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

Because of the special "μ" we picked, we may substitute d{mu}/dx for mu p(x), simplifying the equation to:

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

Using the product rule in reverse, we get:

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

Integrating both sides:

:mu{y} = left(intmu q(x), dx ight) + C

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

:y = frac{left(intmu q(x), dx ight) + C}{mu}

Since "μ" is a function of "x", we cannot simplify any further directly.

A simple example

Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension/compression of the spring. For now, we may 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):

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

If we look for solutions that have the form C e^{kt}, 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:

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

To determine the unknown constants A and B, we need "initial conditions", i.e. equalities that specify the state of the system at a given time (usually t = 0).

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

: x(0) = A cos 0 + B sin 0 = A = 1,

and so A = 1.

: x'(0) = -A sin 0 + B cos 0 = B = 0,

and so B = 0.

Therefore x(t) = cos t. This is an example of simple harmonic motion.

A more complicated model

The above model of an oscillating mass on a spring is plausible but not really realistic. For a start, we have imagined a perpetual motion machine, violating the second law of thermodynamics. Therefore, we will add some friction for realism. Experimental scientists tell us that friction will tend to decelerate the mass and have magnitude proportional to its velocity (i.e. dx/dt). Our new differential equation, expressing the balancing of the acceleration and the forces, is

: 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^{kt}, 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:

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

(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 one would expect a vibrating spring to behave as friction removed the energy from the system.

ee also

* Exact form
* Ordinary differential equation

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.

External links

* [http://eqworld.ipmnet.ru/en/solutions/ode.htm Ordinary Differential Equations] at EqWorld: The World of Mathematical Equations.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Differential equations of addition — In cryptography, differential equations of addition (DEA) are one of the most basic equations related to differential cryptanalysis that mix additions over two different groups (e.g. addition modulo 232 and addition over GF(2)) and where input… …   Wikipedia

  • Examples of vector spaces — This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis. Notation . We will let F denote an arbitrary field such as the real numbers R or the complex numbers C.… …   Wikipedia

  • List of dynamical systems and differential equations topics — This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations. Contents 1 Dynamical systems, in general 2 Abstract dynamical systems 3 …   Wikipedia

  • List of solution strategies for differential equations — Exact= * Method of undetermined coefficients * Integrating factor:For y +a(x)y = b(x) let M(x)=exp{int a(x),dx} then::y(x) = frac{int b(x) M(x), dx + C}{M(x)}., * Method of variation of parameters * Separable differential equationNumerical… …   Wikipedia

  • Spectral theory of ordinary differential equations — In mathematics, the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl… …   Wikipedia

  • List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… …   Wikipedia

  • Differential equation — Not to be confused with Difference equation. Visualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state… …   Wikipedia

  • Differential algebraic equation — In mathematics, differential algebraic equations (DAEs) are a general form of (systems of) differential equations for vector–valued functions x in one independent variable t, where is a vector of dependent variables and the system has as many… …   Wikipedia

  • Differential variational inequality — In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVIs are useful for representing models involving both… …   Wikipedia

  • Differential operator — In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”