Examples of differential equations: Difference between revisions

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==Second-order linear ordinary differential equations==

===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 ''t'' as ''x''(''t''). Now, using [[Newton's laws of motion|Newton's second law]] we can write (using convenient units):

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This is a model of ''[[Damping|''damped oscillator'']]''. The plot of displacement against time would look like this:

: [[Image:Damped Oscillation2.svg|400px|center]]

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can be easily symbolically

[http://www.wolframalpha.com/input/?i=y%27+%3D+{{1%2C2}}%2C{{2%2C-2}}.y%2B+{t%2C+sin%28t%29} solved]

in [[WolframAlpha]].