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: <math>m\frac{d^2x}{dt^2} +kx=0,</math>
where ''m'' is the mass and ''k'' is the spring constant that represents a measure of spring stiffness.
If we look for solutions that have the form <math>Ce^{\lambda t}</math>, where ''C'' is a constant, we discover the relationship <math>\lambda^2+1=0</math>, and thus <math>\lambda</math> must be one of the [[complex number]]s <math>i</math> or <math>-i</math>. Thus, using [[Eulers formula in complex analysis|Euler's theorem]] we can say that the solution must be of the form:
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