**Euler’s Equation**

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The last equation is a homogeneous linear second order differential equation with constant coefficients:

_{}

where

_{}

For these simple equations, we look for a solution of the form

_{}

Substituting these values we get

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Dividing through by _{}gives

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which is the characteristic equation of the differential equation. The solution of this equation can be derived from the quadratic equation

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Substituting the values gives

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The two solutions to the differential equation are

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And the complete solution is

_{}

Recalling that

_{}

So the final solution is

_{}

Taking the derivative

_{}

This is Euler’s equation!