How to solve \(\displaystyle{X}_{{\times}}={\left({1}+\delta{\left\lbrace{\left({x}\right)}\right\rbrace}\right)}{X}\) I am trying to

loraliyeruxi

loraliyeruxi

Answered question

2022-03-31

How to solve Xxx=(1+δ{(x)})X
I am trying to find the vibration modes of a string that has a uniform mass density, plus some point mass somewhere attached to it, modelled by an additional Dirac delta function in the mass density. The wave equation is of the form
uxx=(1+δ(x))utt
where u is the deformation, and (1+δ(x)) the mass density. After separation of variables we find
Xxx=(1+δ(x))X
where X is the spatial part of the solution. Is there any analytical solution for X?

Answer & Explanation

davane6a7a

davane6a7a

Beginner2022-04-01Added 8 answers

Step 1
We want to solve the differential equation y(x)=(1+δ(x))y(x), where δ is the Dirac delta "function".
Since f(x)δ(x)=f(0)δ(x) for every f continuous at x=0, the equation reduces to y(x)=y(x)+y(0)δ(x), i.e.
y(x)y(x)=y(0)δ(x). A solution to this is continuous, satisfies yy=0 on (,0) and on (0,), and its derivative makes a jump at x=0 which is equal to y(0).
Step 2
Let y(x)=Aex+Bex be the solution on (,0) and y(x)=Cex+Dex be the solution on (0,). Then, for continuity we shall have A+B=C+D=:E, and to get yy=y(0)δ we shall have (CD)(AB)=y(0)=E.
Now you only have to find all A,B,C,DR such that A+B=C+D=(CD)(AB).

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