For example, can we show that for first order linear differential equations, the equations before and after multiplying with integrating factor are equivalent? How? That is, the solution of one is the solution of other and vice versa.
The answer is no.
If you multiply a differential equation by a function, you could obtain a new equation which has more solutions if the function you multiply by can be zero on intervals.
By in first order ODE, the integrating factor is an exponential which is non-zero on your domain. To see that this leads to no extra solutions, you can argue by multiplication by the integrating factor (and then by its reciprocal) that every solution if the original equation is a solution of the new equation ( and conversely).