Final answer:
The initial value problem involving the homogeneous differential equation (x + ye^y/x) dx - xe^y/x dy = 0, with y(1) = 0, can be solved using substitution and separation of variables. Integration provides an implicit solution which can be expressed in terms of y and x, respecting the initial condition provided.
Step-by-step explanation:
To solve the given initial value problem, we first observe that the differential equation is homogeneous. The equation provided is (x + yey/x) dx - xey/x dy = 0, with the initial condition y(1) = 0. To solve this, we can try separating variables or applying the method of substitution.
Given that this differential equation is not readily separable, the substitution method seems apt. If we let v = y/x, then y = xv and dy = x dv + v dx. Substituting these into the original equation gives us a separable equation in terms of x and v, which can then be integrated.
After integrating and finding an implicit solution for v, we can then substitute back in terms of y and x to recover an implicit solution for y in terms of x, which satisfies the initial condition y(1) = 0.