Representing differential equations explicitly and implicitly
For the past two weeks I have been dealing with differential equations and I have stumbled upon a seemingly simple problem. Suppose I have a separable differential equation . This simply tells that the function value has to equal to the slope at a particular value. The solution is therefore simply . So far so good. Now suppose I have similiar equation, namely . Now this tells me that the slope of the function at a particular has to equal . Hence, the solution is simply the integral, namely . One does not even need the conventional methods to solve separable differential equations.
However, let me now define . The two differential equations (and hence the solutions) should be now equivalent. As we have seen, that's not the case. I can sense that I am commiting a mathematical fallacy, however, I do not know what seems to be the problem when I solve the differential equation implicitly rather than explicitly as stated above. Could anyone help me please?