Examples of differential equations: Difference between revisions

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===A simple example===

Suppose a mass is attached to a spring which exerts an attractive force on the mass [[Proportionality (mathematics)|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 ''{{mvar|t''}} as {{math|''x''(''t'')}}. Now, using [[Newton's laws of motion|Newton's second law]] we can write (using convenient units):

: <math>m\frac{d^2x}{dt^2} + kx = 0,</math>

where ''{{mvar|m''}} is the mass and ''{{mvar|k''}} is the spring constant that represents a measure of spring stiffness. For simplicity's sake, let us take {{math|1=''m'' = ''k''}} as an example.

If we look for solutions that have the form <{{math>|''Ce^{\lambda ''<sup>λ''t}''</mathsup>}}, where ''{{mvar|C''}} is a constant, we discover the relationship <{{math|1=λ^{\lambda^2} + 1 = 0</math>}}, and thus <{{math>\lambda</math>|λ}} must be one of the [[complex number]]s <math>{{mvar|i</math>}} or <{{math>-|−''i</math>''}}. Thus, using [[Euler's formula]] we can say that the solution must be of the form:

: <math>x(t) = A \cos t + B \sin t.</math>

See a [http://www.wolframalpha.com/input/?i=x%27%27%3D-x solution] by [[WolframAlpha]].

To determine the unknown constants ''{{mvar|A''}} and ''{{mvar|B''}}, we need ''initial conditions'', i.e. equalities that specify the state of the system at a given time (usually&nbsp;{{math|1=''t''&nbsp; =&nbsp; 0}}).

For example, if we suppose at {{math|1=''t''&nbsp; =&nbsp; 0}} the extension is a unit distance ({{math|1=''x''&nbsp; =&nbsp; 1}}), and the particle is not moving ({{math|1=''dxx''/''dt''&nbsp;′ =&nbsp; 0}}). We have

: <math>x(0) = A \cos 0 + B \sin 0 = A = 1, </math>

and so&nbsp;{{math|1=''A''&nbsp; =&nbsp; 1}}.

: <math>x'(0) = -A \sin 0 + B \cos 0 = B = 0, </math>

and so {{math|1=''B''&nbsp; =&nbsp; 0}}.

Therefore {{math|1=''x''(''t'')&nbsp; =&nbsp; cos&nbsp; ''t''}}. This is an example of [[simple harmonic motion]].

See a [http://www.wolframalpha.com/input/?i=x%27%27%3D-x%2Cx%280%29%3D1%2Cx%27%280%29%3D0 solution] by [[Wolfram Alpha]].