seeking for help: too few initial conditions for differential equations in mathcad

I'd say rather than too few initial conditions, you have too many (namely 4 instead of 2).

First of all let's establish that you have a single differential equation for the function W(r). That is because S(r) is NOT involved in the first equation.

The second equation tells that W(r) is the derivative of S(r) multiplied with a constant.

So you only have to solve the first differential equation to find W(r) and then S(r) can be found by integrating.

You might indeed say they're consecutive: one follows the other. They're not simultaneous!

Let's do some work:

I would guess that this solution implicitly assumes that  W(0)=0 and S(0)=0, which is pretty close to two of the initial conditions you've provided.

You say that r ranges from 0 to 0.001. Well here's the plot:

Does this meet your expectations?

Luc

Or, avoiding all the symbolic math,

Here is a third approach, following more closely what you had tried to do:

1) You have two functions W and S - so you have to solve for both, not just for S

2) You had a typo in the last W(r). You had a multiplication W*(r)

3) You cannot chose r=0 as start value -> division by zero (first equation, if solved for the derivation of W).

4) Its was already said: you had too much initial conditions for a system of just two DE's of 1st order

Regards, Werner