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Differential Equations Formulas

Browse 1 reviewed differential equations formulas with concrete inputs, outputs, calculations, applicability checks, and LaTeX.

Differential Equations

Laplace Transform

F(s)=\int_0^{\infty}f(t)e^{-st}\,dt

Maps a time-domain function to a complex-frequency function useful for initial-value problems.

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Differential Equations

Exponential Decay

N(t)=N_0e^{-\lambda t}

Models decay at a constant proportional rate.

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Differential Equations

Exponential Decay Model

y(t)=y_0e^{-kt}

Solves a proportional decay law y′=−ky with positive rate k.

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Differential Equations

First-Order Linear ODE Solution

y=e^{-\int P(x)dx}\left(\int Q(x)e^{\int P(x)dx}dx+C\right)

Gives the integrating-factor solution of y′+P(x)y=Q(x).

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Differential Equations

Heat Equation

\frac{\partial u}{\partial t}=\alpha\nabla^2u

Models diffusion of heat or another conserved quantity through a medium.

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Differential Equations

Laplace Equation

\nabla^2u=0

Defines harmonic functions and steady-state potential fields.

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Differential Equations

Logistic Differential Equation

\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)

Models growth that slows as a population approaches carrying capacity.

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Differential Equations

Logistic Growth Solution

N(t)=\frac{K}{1+Ae^{-rt}}

Describes bounded growth toward a carrying capacity.

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Differential Equations

Simple Harmonic Oscillator Equation

m\ddot x+kx=0

Describes an ideal mass-spring oscillator without damping or forcing.

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Differential Equations

Wave Equation

\frac{\partial^2u}{\partial t^2}=c^2\nabla^2u

Models ideal wave propagation with constant speed c.

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