Examples of differential equations: Difference between revisions

Revision as of 14:37, 8 February 2021

Revision as of 14:44, 8 February 2021

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===A more complicated model===

The above model of an oscillating mass on a spring is plausible but not very realistic: in practice, [[friction]] will tend to decelerate the mass and have magnitude proportional to its velocity (i.e. {{math|''dx''/''dt''}}). Our new differential equation, expressing the balancing of the acceleration and the forces, is

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

where <math>{{mvar|c</math>}} is the damping coefficient representing friction. Again looking for solutions of the form {{math|''Ce''<mathsup>Ce^{\lambda λ''t}''</mathsup>}}, we find that

: <math>m\lambda^2 + c \lambda + k = 0. </math>

This is a [[quadratic equation]] which we can solve. If <{{math>|''c^''{{sup|2}} <4km</math> 4''km''}} there are two complex conjugate roots {{math|''a''&nbsp; ±&nbsp; ''ibbi''}}, and the solution (with the above boundary conditions) will look like this:

: <math>x(t) = e^{at} \left(\cos bt - \frac{a}{b} \sin bt \right) </math>

Let us for simplicity take <{{math>|1=''m'' = 1</math>}}, then <{{math>|1=0 < ''c'' =-2a</math> −2''a''}}, and <{{math>|1=''k'' = ''a^''{{sup|2}} + ''b^''{{sup|2</math>}}}}.

The equation can be also solved in MATLAB symbolic toolbox as