tra poco vi lascio.......

[Professeur Bugigiò]

io tra poco vi lascio per questo:

For a first-order linear ODE, with coefficients that may or may not vary with x:
y'(x) + p(x)y(x) = r(x)
Then,
where κ is the constant of integration, and

This proof comes from Jean Bernoulli. Let
Suppose for some unknown functions u(x) and v(x) that y = uv.
Then
Substituting into the differential equation,
Now, the most important step: Since the differential equation is linear we can split this into two independent equations and write
Since v is not zero, the top equation becomes
The solution of this is
Substituting into the second equation
Since y = uv, for arbitrary constant C

As an illustrative example, consider a first order differential equation with constant coefficients:
This equation is particularly relevant to first order systems such as RC circuits and mass-damper systems.
In this case, p(x) = b, r(x) = 1.
Hence its solution is

Ah, le equazioni differenziali..... Tutt le volte che ne incontro una devo andare a rvedermi come si risolvono.