A second order non-homogeneous differential equation is of the form y"+a.y'+b.y=g(x). For non-homogeneous differential equation g(x) must be non-zero. A second order Euler-Cauchy differential equation x^2 y"+ a.x.y'+b.y=g(x) is called homogeneous linear differential equation, even g(x) may be non-zero. Also, after the substitution x=exponential (z) to obtained the solution, Euler-Cauchy differential equation is still non-homogeneous. Then, why it is called homogeneous linear, whereas it comes under the non-homogeneous differential equation?
Dear Pankaj
A second order Euler-Cauchy differential equation x^2 y"+ a.x.y'+b.y=g(x) is called
non-homogeneous linear differential equation if g(x) is non-zero
Read Attached File
and
Watch the video on the following link
https://www.youtube.com/watch?v=_-Wh5Sopd_M
very thanks to you sir for the link, but in almost all engineering books what I have studied, the Euler-Cauchy Diff. Equation is named as Homogeneous Linear Diff. Equation. Is there need a correction.
If g(0)=0, then Y=0 is a solution of Euler-Cauchy Equation. In this case we can say (formally) that the last one is homogeneous.
I am teaching in Engineering, yes I observed that some books of Engineering mathematics used what you said but I think, it is not the good practice....
Here homogeneous in the sense that degree of the variable coefficient and the order of the derivative are equal in their respective terms (like x^2 D^2, xD etc.,), irrespective of the RHS.This is my opinion.
Thanks to you dear Mahesh sir. I have also one question asked about the degree of an ordinary differential equation. I request you to look and give answer/suggestions.
@ Rajaraman sir
I am not agree to you sir because the sense of a homogeneous equation is the same in all respective definitions.
Each equation is said to be homogeneous if with its each solution y the \lambda y is its a solution too for arbitrary constant \lambda. So when g(0) is null the last property for Euler-Cauchy Equation.hols.
Since the solution is a power of x. Like y=x^n
As hen you take dy/dx it looses one degree (as dy/dx=n x^(n-1))
So by multiplying x to it, we have x(dy/dx) which also has degree n.
Similarly, d^2y/dx^2 it looses two degrees (as d^2y/dx^2=n(n-1) x^(n-2))
But by multiplying x^2 to it, we have x^2(d^2y/dx^2) which also has degree n.
And so on. Thus Cauchy-Euler equation is called homogeneous equation.
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