Eulerís Equation



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





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!