# How to find the auxillary equation and the final solution for #(d^2Phi)/(dphi^2) + BPhi = 0# assuming #Phi = e^(im_lphi)#?

How to find the auxiliary equation and the final solution

for

We have:

This is a second order linear Homogeneous Differentiation Equation with constant coefficients. The standard approach is to find a solution of the homogeneous equation by looking at the Auxiliary Equation, which is the polynomial equation with the coefficients of the derivatives.

Complementary Function

The associated Auxiliary equation is:

The sign of B will determine the possible solution. Then

Here, we are given the form of the solution. Let us consider the given solution:

Using Euler's formula , we can write this given solution as:

We further conclude that:

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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