# I'm stuck with a differential equation

• vinodjoshi

#### vinodjoshi

Hello ppl
I stuck with a differential equation and all I need is your kind help. While solving a problem I got one eqaution as given below

((Ax+B)^4)y''''+ ((Ax+B)^2)y''+(i/const.)y=0

here A, B, and const are constants

i denotes complex no (iota).

I need your kind help. Please suggest me the possible way to solve this equation.

First, change variable

x = (tA-B)/A,

where t is new variable. Your ODE in new variable is

(At)^4y''''+ (At)^2y''+(i/const.)y=0 (2)

this is an "equipotential" equation, also known as an "Euler-type equation".

Then seek solution in form y(t)=t^z. Substitution y(t)=t^z to (2) leads to

t^z(A^4 z^4-6 A^4 z^3+11 A^4 z^2-6 A^4 z+A^2 z^2-A^2 z+(i/const.))=0,

There are four solutuons of algebraic equation (z1,z2,z3,z4)

A^4 z^4-6 A^4 z^3+11 A^4 z^2-6 A^4 z+A^2 z^2-A^2 z+(i/const.)=0,

so the general solution of ode (2) is as follows

y(t)=C1t^z1+...+C4t^z4

where C1...C4 are arbitrary constants.

Thanks kosovtsov for your kind help.

Sir can you briefly explain about these "Euler Type Equations". Can you give me some references for these equations.

Sir
Can we treat i/const. as a constant. I am confused about it. Please help me in this regard. Can you suggest me some book having similar kind of problem. I will be thankful to you.

Can we treat i/const. as a constant.

Yes, I have considered i/const as a constant. But real meaning of it belong exclusively to the physical problem setting. I know nothing on the physical base of your problem.

Sir, The physical problem belongs to planer wave where the physical medium which carries this wave have tapered cross section. Can this thing help you or I have to explain the problem in detail. All I want to ask (i/const.) is a complex no. so how can we take it as real no.