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Static vs dynamic mathematical model

Short answer

Static models describe a system at a fixed instant or equilibrium without time, e.g., Ohm's law V=IR. Dynamic models incorporate time, using differential/difference equations to describe state evolution, e.g., population growth dP/dt=kP. Static models give instantaneous relationships; dynamic models capture transient behaviour and depend on history.

Differentiate between static mathematical model and dynamic mathematical model with suitable examples.

A static mathematical model describes the relationship between variables at a fixed moment or in equilibrium; time does not appear in it. A dynamic mathematical model describes how the variables change with time, normally using differential or difference equations.

PointStatic modelDynamic model
TimeNot a variableTime is an independent variable
FormAlgebraic equationsDifferential or difference equations
DescribesSystem at one instant or steady stateSystem behaviour over time, transient and steady
OutputDepends only on the present inputDepends on present input and past history/state
MemoryNoneHas memory (state variables)
SolutionDirect substitution or algebraAnalytical integration or numerical methods, simulation
ComputationSimpleMore complex
UseDesign, equilibrium and sizing studiesPrediction, stability and response studies

Examples of static models

  • Ohm's law, V = I R: the voltage depends on the present current only.
  • Straight-line cost model: C = F + v x, where F is the fixed cost, v the unit cost and x the quantity.
  • Economic order quantity: Q = sqrt(2 D S / H).
  • Population of a town estimated by P = density x area.

Examples of dynamic models

  • Population growth: dP/dt = k P, solution P = P0 e^(kt).
  • Spring-mass-damper (car suspension): M x'' + D x' + K x = F(t).
  • RC circuit: R C dv/dt + v = V(t), whose capacitor voltage changes with time.
  • Inventory over days: I(n+1) = I(n) + production(n) - demand(n) (difference equation, a discrete dynamic model).

A dynamic model reduces to a static one when all derivatives are set to zero (steady state). For example, setting dv/dt = 0 in the RC circuit gives v = V, a static relationship.

✓ Answer checked by Claude Opus 5.5 (high effort)

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