Initial Value Problem Or Ivp

Find the implicit IVP solution

y'(x) = – (ex ⁄ ey+yey)

y(0) = 0

What can we say about the existence of the explicit solution? If there is, can we say who it is?

Solution

SEparate variables: (ey + yey)dy = – exdx;

We integrate now both sides: ∫(ey + yey)dy = – ∫exdx; ey + yey – ey = – ex + C; yey = – ex + C.

y(0) = 0; 0·e0 = – 1 + C; C = 0.

So, yey = – ex + 1 is implicit solution of this IVP