If I have the following differential equation
$$ dx/dt=x $$
The ‘normal’ way of solving a differential equation is with separation of variables
$$ dx/x=dt $$
Integrating both sides you get
$$ ln(x)=t + C $$
Raising to the power of e
$$ x(t)=e^t+e^C $$
Why isn’t this equally as valid
$$ dx/dt=x $$
Integrate both sides with respect to t
$$ x=xt+C $$
Which obviously doesn’t match. But also another possible solution would be
$$ x $$
Taking the derivative you get
$$ dx/dt = x $$
What am I missing? Is this just a matter of me misunderstanding the syntax? The other thing I notice is with separation of variables the left side is integrated with respect to x but right hand with respect to y.