\textbf{Linear differential equation of the first order:}

,,\qquad \boxed{y' + Py = Q}

\textbf{General solution:}
Multiply both sides with \[e^{\int P\mathrm{d}x}\]

,,\qquad y'e^{\int P\mathrm{d}x} + Pye^{\int P\mathrm{d}x} = Qe^{\int P\mathrm{d}x}

Collapse left side \[(fg)' = f'g + fg'\]

,,\qquad \left(ye^{\int P\mathrm{d}x}\right)' = Qe^{\int P\mathrm{d}x}

Integrate both sides

,,\qquad ye^{\int P\mathrm{d}x} = C + \int Qe^{\int P\mathrm{d}x}\mathrm{d}x

Multiply both sides with \[e^{-\int P\mathrm{d}x}\]

,,\qquad \boxed{y = e^{-\int P\mathrm{d}x}\left(C + \int Qe^{\int P\mathrm{d}x}\mathrm{d}x \right)}


\textbf{Example:}

,,\qquad \boxed{y' + 2xy = 4x}

\textbf{Solution:}
Since \[P = 2x,\enspace Q=4x\]:
,,\qquad y = e^{-x^2}\left(C  + \int 4xe^{x^2}\mathrm{d}x\right) = e^{-x^2}(C + 2e^{x^2}) = \boxed{Ce^{-x^2}  + 2}