Differential Equations
Essential formulas and concepts in differential equations
Exponential Growth/Decay
\(\displaystyle \frac{dy}{dt} = k\,y \quad\implies\quad y(t) = C\,e^{k t}\)
Variables:
- \(y(t) is the quantity as a function of time t, k is the growth (k>0) or decay (k<0) rate constant, C is the initial amount y(0).\)
Description/Usage:
The differential equation states that the rate of change of \(y\) is proportional to its current amount. The solution is an exponential law. Examples: radioactive decay (negative \(k\)), population growth (positive \(k\)). The graph is an exponential curve increasing or decreasing over time.
Mathematical concept visualization
Simple Harmonic Motion (solution)
\(x'' + \omega^2 x = 0 \quad\implies\quad x(t) = A \cos(\omega t) + B \sin(\omega t)\)
Variables:
Description/Usage:
Mathematical concept visualization