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First-order differential equation

A separable differential equation $y'=ky$ has solution $y=Ce^{kx}$ , where $C$ is determined by an initial condition. For a more general linear equation $y'+p(x)y=q(x)$ we use an integrating factor.

y' = ky\ \Rightarrow\ y = Ce^{kx}

separable equation, exponential growth/decay

\mu(x) = e^{\int p(x)\,dx}

integrating factor for

y'+p(x)y=q(x)

Symbols

$k$	growth rate (constant)
$C$	constant, determined by $y(0)$

Example

$y'=-2y$ , $y(0)=5$ : $y=5e^{-2x}$ , so $y(\ln 2)=5\cdot2^{-2}=1.25$ .

Multiplying the equation by the integrating factor

\mu

turns the left side into

(\mu y)'

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Part of Linear Algebra and Differential Equations: Differential equations.