Solution formula for first-order linear equations

Property — General solution

The equation $y' + p(x)\,y = q(x)$ has general solution: $y(x) = e^{-P(x)}\left[\int q(x)\,e^{P(x)}\,dx + C\right], \qquad P(x)=\int p(x)\,dx.$

Two cases are distinguished:

• homogeneous case ($q=0$): gives $y = Ce^{-P(x)}$;
• non-homogeneous case ($q\ne 0$): adds a particular solution to the homogeneous solution.

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Topics: Differential equations
Concepts: First-order linear ODE
Skills: Integrating · Solving equations · Using formulae