Eulerís Equation


 

 

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

 

 

 

Dividing through by gives

 

 

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

 

 

Substituting the values gives

 

 

The two solutions to the differential equation are

 

 

And the complete solution is

 

 

Recalling that

 

 


So the final solution is

 

 

Taking the derivative

This is Eulerís equation!