Let us consider
sol1 := dsolve({diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1],
y(0) = 1}, numeric);
sol1 := proc(x_rkf45) ... end proc
The problem under consideration has the symbolic solution:
sol2 := dsolve({diff(y(x), x) = solve((1/2)*(diff(y(x), x))^2 = (1-ln(y(x)^2))*y(x)^2, diff(y(x), x))[1],
y(0) = 1});
sol2 := y(x) = exp(x*sqrt(2)-x^2)
Let us compare the plots of sol1 and sol2 (which should coincide):
A := plots:-odeplot(sol1, x = 0 .. 1, color = navy, style = point):
B := plot(rhs(sol2), x = 0 .. 1, color = red):
plots:-display([A, B]);
The plots differ after approximately 0.707. Bug_in_dsolve_numeric.mw
Edit. The title and one of the tags.