Differential Equations

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Formulate a simple statement involving a rate of change as a differential equation (including introducing and evaluating a constant of proportionality where necessary)

  2. Find by integration a general form of solution for a first order differential equation in which the variables are separable (using any integration techniques you've learned)

  3. Use an initial condition to find a particular solution to a differential equation

  4. Interpret the solution of a differential equation in the context of a problem being modelled by the equation

What is a Differential Equation?

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A differential equation is simply an equation that contains at least one derivative (a term like dy/dx or dv/dt).

Why do we use derivatives? Remember that derivatives represent rates of change. For example:

  • dy/dx means "the rate at which y changes as x changes"
  • dv/dt means "the rate at which velocity v changes as time t changes"

Examples of Differential Equations

First order differential equations contain only first derivatives (like dy/dx, dv/dt):

  • dv/dt = 3t
  • dy/dx = 2x + 3

Second order differential equations contain second derivatives (like d²y/dx²):

  • d²y/dx² - 3 = 2x
  • d²A/dr² + 5(dA/dr) + 6 = 0

In this course, we only study first order differential equations.

General Solutions

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What happens when we solve a differential equation?

When we solve a differential equation, we use integration to work backwards from the derivative to the original function.

When we integrate, we always get a constant of integration (we call it c).

For example, if we solve the differential equation:

  • dv/dt = 2

We integrate both sides:

  • v = ∫ 2 dt
  • v = 2t + c

What is a general solution?

Because we have this constant c, there are actually infinitely many possible solutions to the differential equation. Each different value of c gives us a different solution.

These are called a family of solutions, and we call v = 2t + c the general solution.

Visual example: If you have the differential equation dy/dx = 4x, the general solution is y = 2x² + c. This represents a family of parabolas, all the same shape, but shifted up or down depending on the value of c:

  • When c = 4: y = 2x² + 4
  • When c = 0: y = 2x²
  • When c = -2: y = 2x² - 2

All of these are valid solutions to the same differential equation!

Example: Finding a General Solution

Find the general solution of: 4 - dv/dt = 2

Solution:

  1. Rearrange to get the derivative by itself:

    • 4 - dv/dt = 2
    • dv/dt = 2

  2. Integrate both sides with respect to t:

    • v = ∫ 2 dt
    • v = 2t + c

Answer: v = 2t + c (this is the general solution because c is unknown)

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