Delay Differential Equations with variable delay

Questions
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1

Since 2t = t + t, the following is a delay differential equation with variable delay -t, hence currently unsolvable by Mathematica.

In[44]:= 
NDSolve[{2 y'[t] == x'[t], y[2 t] == 2 x[t], y[t /; t <= 0] == 0,
  x[t /; t <= 0] == 0}, {y, x}, {t, 0, 4}]

During evaluation of In[44]:= NDSolve::cdelay: The method currently implemented for delay differential equations does not support delays that depend directly on the time variable or dependent variables.

There must be some way that Mathematica could be used to solve simple systems like this. I would really appreciate it if someone could help me out here.

Richard Jensen

2

You can see that any multiple of a solution is also a solution. Unless the only solution is zero, the problem has infinitely many solutions, and NDSolve is not appropriate for the task. One nonzero solution is found easily: y[t_] = t^2, x[t_] = 2 t^2 for t>0. Here is a derivation that uses Mathematica.

The equation y[2 t] == 2 x[t] means that x == Function[t, y[2 t]/2]. Let us replace into the other equation:

2  y'[t] == x'[t] /. x -> Function[t, y[2 t]/2]

which gives

2 y'[t] == y'[2 t]

A sufficient condition for this relation is that y' is a monomial of the first degree, for example of the form 2 t. Integrate this and you get y[t_] = t^2. Go back and you find the corresponding x.

3

You are very helpful. Thank you sincerely.