I'm stuck with a differential equation

[vinodjoshi]

Please Help ...... Urgent

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.

Thanks in advance

 

[kosovtsov]

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.

 

[vinodjoshi]

Thanks kosovtsov for your kind help.

 

[vinodjoshi]

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

 

[kosovtsov]

See

http://eqworld.ipmnet.ru/en/solutions/ode/ode0416.pdf

and references therein.

 

[vinodjoshi]

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.

 

[kosovtsov]

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.

 

[vinodjoshi]

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.