---
title: "Dynamic mathematical model with example"
subject: "Simulation and Modeling"
chapter: 2
key: "semester_8/simulation_and_modeling:dynamic-mathematical-model"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/dynamic-mathematical-model
site: BCT Engineering
---

# Dynamic mathematical model with example

Simulation and Modeling · Chapter 2: Physical and Mathematical models

- Asked in: 2072 Magh, 6 marks; 2070 Bhadra, 6 marks (asked 2 times)
- Video: [Math Modeling: Dynamic Systems](https://www.youtube.com/watch?v=VBecZbh3WMg) — Rajendra Dahal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/dynamic-mathematical-model
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-2

## Question

What is a dynamic mathematical model? Explain with examples.

## Short answer

A dynamic mathematical model uses differential or difference equations with time as a variable, requiring initial conditions. Examples: population growth dP/dt = kP gives P = P0 e^(kt); mass-spring-damper M x'' + D x' + K x = F(t).

## Answer

A **dynamic mathematical model** represents a system with equations in which time is a variable. The state of the system at any time depends on its earlier states. It is written as differential equations for continuous systems or difference equations for discrete-time systems, and is solved analytically or numerically (simulation).

General form with state vector `x(t)`, input `u(t)`:

`dx/dt = f(x, u, t)`

#### Characteristics
- Contains derivatives (or time-step differences) of the variables.
- Needs initial conditions, e.g. `x(0) = x0`.
- Output depends on both the input and the history of the system.
- Can be linear or nonlinear, deterministic or stochastic.

#### Example 1: population growth
If the growth rate is proportional to the population P, `dP/dt = k P`, `P(0) = P0`. The solution is `P(t) = P0 e^(kt)`. For k = 0.02 per year, P0 = 10,000, after 10 years `P = 10,000 e^0.2 = 12,214`.

#### Example 2: mass-spring-damper
`M x'' + D x' + K x = F(t)`. This second-order equation models vibration of a vehicle suspension. Setting `x1 = x`, `x2 = x'` gives the state form `x1' = x2`, `x2' = (F - D x2 - K x1)/M`.

#### Example 3: cooling of a body
`dT/dt = -k (T - Ta)`. T falls exponentially towards ambient temperature Ta.

#### Example 4: discrete form
Bank balance: `B(n+1) = B(n) (1 + r) + d(n)` (difference equation, r = interest rate, d = deposit).

#### Solution methods
- Analytical (integration) when the equation is simple.
- Numerical integration (Euler: `x(t+h) = x(t) + h f(x,t)`, Runge-Kutta) on a digital computer.
- Analog computers, which solve the equations with integrators and summers.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A dynamic mathematical model of a discrete-time system is written as:
   A. An algebraic equation
   B. A difference equation
   C. A truth table
   D. A single constant
   Answer: B. Continuous systems use differential equations; discrete-time systems use difference equations. (Key point: Dynamic model form)

2. For dP/dt = kP with k = 0.02 per year and P0 = 10,000, the population after 10 years is about:
   A. 10,200
   B. 12,000
   C. 12,214
   D. 20,000
   Answer: C. P = P0 e^(kt) = 10,000 e^0.2 = 12,214. (Key point: Population growth calculation)

3. Which is the Euler method update for dx/dt = f(x,t)?
   A. x(t+h) = h f(x,t)
   B. x(t+h) = x(t) f(x,t)
   C. x(t+h) = x(t) - h
   D. x(t+h) = x(t) + h f(x,t)
   Answer: D. Euler's method advances the state by step h times the derivative f(x,t). (Key point: Numerical solution)

## Related in Simulation and Modeling

- [Static vs dynamic mathematical model](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Astatic-vs-dynamic-math-model) — Simulation and Modeling, chapter 2, asked 3×
- [Static mathematical model with example](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Astatic-mathematical-model) — Simulation and Modeling, chapter 2
- [Dynamic physical model with example](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Adynamic-physical-model) — Simulation and Modeling, chapter 2, asked 3×

## Related in other subjects

- [Dynamic model of armature-controlled DC motor](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Adc-motor-de-dynamic-model) — Control Systems, chapter 2
- [Dynamic response of excitation system](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-dynamic-response) — Power Plant Equipment, chapter 3, asked 3×
- [Need for excitation and improving dynamic response](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_plant_equipment%3Aexcitation-dynamic-improvement) — Power Plant Equipment, chapter 3