Strange initial values...

[AlfredFlaßhaar]

For some time now, I've been interested in ordinary differential equations of the following type: x*y´´ + y = 0 for the (incomplete) initial value y(0) = 1. These differential equations in the complex plane are discussed extensively in the literature (e.g., Schlömilch, Smirnov 3/2,...) – closed-form solutions exist only using well-known transcendental functions or in series form. Now, however, this linear equation must be considered in the real realm. Is there a way to construct solutions for a sequence of initial values ​​(eps;1) with eps-->0 and to create plots close to the initial value? The second initial value for y´(0) can be chosen "appropriately." My MC14 knowledge is insufficient for this.

[LucMeekes]

Do you mean that y is a function of x, or are y and x both functions (of t, e.g.)?

Luc

[AlfredFlaßhaar]

It is assumed that y = y(x). This is also indicated by the incomplete initial value y(0) = 0. Perhaps the problem can be solved by substituting y' = z and the resulting first-order linear system? For eps --> 0, the "appropriate" initial value for y'(0) must be kept constant.

[LucMeekes · Accepted solution]

Hi Alfred,

The general solution to the DE is:

Its derivative is:

(And the second derivative is:

so that:

and that's the negative of y(x). which proves that y(x) shown above is a/the solution to the DE.)

Let's guess that C₀=1 and C₁=0 and plot the function and its derivative:

Seems like y(0)=0 and y'(0) can be chosen to be 1.

Simple evaluation of y(0,C) results in a singularity error, but a limit does the job:

Hope this helps.

Success!
Luc

[AlfredFlaßhaar]

Thank you, perfect ;-). The initial value y(0) = 1 is therefore not realizable by the general solution.