Linear First Order ODE/x y' + y = f (x)

Theorem

The linear first order ODE:

$(1): \quad x \, \dfrac {\d y} {\d x} + y = \map f x$

has the general solution:

$\ds x y = \int \map f x \rd x + C$

Proof

\(\ds x \dfrac {\d y} {\d x} + y\)

\(=\)

\(\ds \map f x\)

\(\ds \leadsto \ \ \)

\(\ds x \rd y + y \rd x\)

\(=\)

\(\ds \map f x \rd x\)

\(\ds \leadsto \ \ \)

\(\ds \map \d {x y}\)

\(=\)

\(\ds \map f x \rd x\)

Product Rule for Differentials

\(\ds \leadsto \ \ \)

\(\ds x y\)

\(=\)

\(\ds \int \map f x \rd x + C\)

$\blacksquare$