8.1.3 Modelling using First Order Differential Equations

Introduction to Modelling with Differential Equations

First-order differential equations are used extensively to model real-world phenomena in contexts such as kinematics, population growth, and chemical reactions. A differential equation models the relationship between a rate of change and the variables involved.

Common Scenarios for Modelling

Kinematics

In kinematics, differential equations often relate velocity, acceleration, and displacement.

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Example Relationship: If $v = \frac{ds}{dt}$ and $a = \frac{dv}{dt}$ , we can express acceleration as:

a = v \frac{dv}{ds}

Population Growth or Decay

Population growth can be modelled using the equation:

\frac{dP}{dt} = kP

where $k$ is the growth rate. This results in exponential growth or decay depending on the sign of $k$ .

Newton's Law of Cooling

The rate of change of temperature $T$ of an object is proportional to the difference between its temperature and the ambient temperature $T_a$ :

\frac{dT}{dt} = -k(T - T_a)

Motion with Resistance

The motion of an object experiencing resistance proportional to velocity can be modelled as:

m \frac{dv}{dt} = -kv

where $k$ is the resistance constant.

Worked Examples

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Example 1: Modelling Kinematics with Resistance

An object of mass mm falls under gravity with air resistance proportional to its velocity $v$ . Model and solve for $v(t)$ .

Step 1: Write the Equation

The forces acting are gravity ( $mg$ ) and air resistance ( $kv$ ):

m \frac{dv}{dt} = mg - kv

Rearrange:

\frac{dv}{dt} = g - \frac{k}{m}v

Step 2: Solve the Differential Equation

This is a linear first-order differential equation:

\frac{dv}{dt} + \frac{k}{m}v = g

Find the integrating factor:

F(t) = e^{\int \frac{k}{m} \, dt} = e^{\frac{k}{m}t}

Multiply through by $F(t)$ :

e^{\frac{k}{m}t} \frac{dv}{dt} + \frac{k}{m}e^{\frac{k}{m}t}v = ge^{\frac{k}{m}t}

Recognize as a product rule:

\frac{d}{dt}\left(e^{\frac{k}{m}t}v\right) = ge^{\frac{k}{m}t}

Integrate both sides:

e^{\frac{k}{m}t}v = \int ge^{\frac{k}{m}t} \, dt = \frac{mg}{k}e^{\frac{k}{m}t} + C

Solve for $v$ :

v = \frac{mg}{k} + Ce^{-\frac{k}{m}t}

Step 3: Interpret the Solution

As $t \to \infty$ , $Ce^{-\frac{k}{m}t} \to 0$ , so $v \to \frac{mg}{k}$ .

This is the terminal velocity.

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Example 2: Population Growth

Model and solve a population growth scenario where the growth rate is proportional to the population, starting with $P(0) = P_0$

Step 1: Write the Equation

\frac{dP}{dt} = kP

Step 2: Solve the Differential Equation

This is separable:

\frac{1}{P} \, dP = k \, dt

Integrate both sides:

\ln |P| = kt + C

Exponentiate:

P = e^{kt+C} = Ce^{kt}

Use the initial condition $P(0) = P_0$

P_0 = Ce^0 \implies C = P_0

Solution:

P(t) = P_0 e^{kt}

Step 3: Interpret the Solution

If $k>0$ the population grows exponentially.

If $k < 0$ the population decays exponentially.

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Example 3: Newton's Law of Cooling

A hot object cools in a room at temperature $T_a = 20^\circ C$

Initially, the object's temperature is $T(0) = 100^\circ C$

Solve for $T(t)$

Step 1: Write the Equation

\frac{dT}{dt} = -k(T - T_a)

Substitute $T_a = 20$

\frac{dT}{dt} = -k(T - 20)

Step 2: Solve the Differential Equation

This is separable:

\frac{1}{T - 20} \, dT = -k \, dt

Integrate:

\ln|T - 20| = -kt + C

Exponentiate:

T - 20 = Ce^{-kt}

Solve for $T$

T = 20 + Ce^{-kt}

Step 3: Apply the Initial Condition

When $t = 0, T(0) = 100$

100 = 20 + C \implies C = 80

Solution:

T(t) = 20 + 80e^{-kt}

Note Summary

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Common Mistakes:

Incorrectly separating variables: Ensure terms involving $t$ and $y$ are properly isolated before integrating.
Forgetting the constant of integration ( $C$ ): Always include $C$ after integration.
Mismanaging initial conditions: Check that initial conditions are used correctly to determine constants.
Incorrect signs in exponential models: Be cautious with negative exponents, especially in cooling or decay problems.
Mishandling proportional relationships: Ensure constants like $k$ have correct interpretations (e.g., growth or decay rates).

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Key Formulas:

Population Growth/Decay:

\frac{dP}{dt} = kP \quad \implies \quad P(t) = P_0 e^{kt}

Newton's Law of Cooling:

\frac{dT}{dt} = -k(T - T_a) \quad \implies \quad T(t) = T_a + Ce^{-kt}

Motion Under Gravity with Resistance:

m \frac{dv}{dt} = mg - kv \quad \implies \quad v(t) = \frac{mg}{k} + Ce^{-\frac{k}{m}t}

Linear Differential Equation (General Form):

\frac{dy}{dx} + P(x)y = Q(x) \quad \implies \quad y = \frac{1}{F(x)} \left[ \int F(x)Q(x) \, dx + C \right]

where $F(x) = e^{\int P(x) \, dx}$ is the integrating factor.

Modelling using First Order Differential Equations Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

8.1.3 Modelling using First Order Differential Equations

Introduction to Modelling with Differential Equations

Common Scenarios for Modelling

Kinematics

Population Growth or Decay

Newton's Law of Cooling

Motion with Resistance

Worked Examples

Example 1: Modelling Kinematics with Resistance

Example 2: Population Growth

Example 3: Newton's Law of Cooling

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Modelling using First Order Differential Equations For their A-Level Exams.

Other Revision Notes related to Modelling using First Order Differential Equations you should explore

Modelling using First Order Differential Equations Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

8.1.3 Modelling using First Order Differential Equations

Introduction to Modelling with Differential Equations

Common Scenarios for Modelling

Kinematics

Population Growth or Decay

Newton's Law of Cooling

Motion with Resistance

Worked Examples

Example 1**: Modelling Kinematics with Resistance**

Example 2**: Population Growth**

Example 3**: Newton's Law of Cooling**

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Modelling using First Order Differential Equations For their A-Level Exams.

Other Revision Notes related to Modelling using First Order Differential Equations you should explore

Example 1: Modelling Kinematics with Resistance

Example 2: Population Growth

Example 3: Newton's Law of Cooling