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

##### 1 Answer

#### Answer:

How to find the auxiliary equation and the final solution

for

#### Explanation:

We have:

# (d^2Phi)/(dphi^2) + BPhi = 0 #

We assume that

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:

# m^2+0m+B = 0#

# :. m^2 = B#

The sign of B will determine the possible solution. Then

**Case 1:**

Real distinct roots, given by

#m=+-B # , so that:

# Phi = C_1e^(Bphi) + C_2Be^(-Bphi) #

**Case 2:**

Pure imaginary roots, given by

#m=+-Bi # , so that:

# Phi = e^(0phi){C_1cos(Bphi) + C_2sin(Bphi)} #

# \ \ \ = C_1cos(Bphi) + C_2sin(Bphi) #

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

# Phi = e^(i m_l phi) #

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

# Phi = cos(m_l phi) + isin(m_l phi) #

Comparing the given solution with the two possible cases we conclude that

# Phi = C_1cos(Bphi) + C_2sin(Bphi) #

We further conclude that:

# C_1=1, C_2=i, B=m_l #