How to solve&nbsp;X×=(1+δ{(x)})XI 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 formuxx=(1+δ(x))uttwhere u is the deformation, and&nbsp;(1+δ(x))&nbsp;the mass density. After separation of variables we findX×=(1+δ{(x)})Xwhere X is the spatial part of the solution. Is there any analytical solution for X?

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How to solve&amp;nbsp;X×=(1+δ{(x)})XI 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 formuxx=(1+δ(x))uttwhere u is the deformation, and&amp;nbsp;(1+δ(x))&amp;nbsp;the mass density. After separation of variables we findX×=(1+δ{(x)})Xwhere X is the spatial part of the solution. Is there any analytical solution for X?

smachttenbem · Accepted Answer

Step 1We want to solve the differential equation y″(x)=(1+δ(x))y(x), where δ is the Dirac delta &quot;function&quot;.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 y″−y=0 on (−∞,0) and on (0,∞), and its derivative makes a jump at x=0 which is equal to y(0).Step 2Let y(x)=Aex+Be−x be the solution on (−∞,0) and y(x)=Cex+De−x be the solution on (0,∞). Then, for continuity we shall have A+B=C+D=:E, and to get y″−y=y(0)δ we shall have (C−D)−(A−B)=y(0)=E.Now you only have to find all A,B,C,D∈R such that A+B=C+D=(C−D)−(A−B).

How to solve X_{xx}=(1+\delta{(x)})X I am trying to find the vibration modes of a string t

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