Ordinary differential equations: first and second order
Differential equations · Physics
Study notes
Consider a simple first-order ODE representing radioactive decay: dy/dt = -ky, where y is the amount of substance and k is the decay constant. To solve this, we separate variables: dy/y = -k dt. Integrating both sides gives ln(y) = -kt + C. Exponentiating both sides yields y = e^(-kt+C) = A e^(-kt), where A = e^C is a constant determined by initial conditions. If y(0) = y0, then A = y0. Thus, y(t) = y0 e^(-kt). This shows the amount of substance decreases exponentially over time.