Introduction to Differential Equations

Welcome to the world of Differential Equations! While the name sounds a bit intimidating, a differential equation is simply any equation that involves a derivative (like \(\frac{dy}{dx}\) or \(\frac{dv}{dt}\)).

In your previous calculus chapters, you learned how to differentiate and integrate. Now, you will use those skills to solve problems where the rate of change of a quantity is related to the quantity itself. This is how scientists model everything from the spread of a virus to the way a cup of tea cools down.

1. Constructing Differential Equations

Before you can solve a differential equation, you often need to "build" it from a description in words. This is a key skill for AO3 (Problem Solving) marks.

Look out for these common phrases:

  • "The rate of change of \(y\) with respect to \(x\)": This means \(\frac{dy}{dx}\).
  • "Is proportional to...": This means you use the symbol \(\propto\), which becomes \(= k\) when you turn it into an equation.
  • "The rate of increase/growth": This implies a positive derivative.
  • "The rate of decrease/decay": This implies a negative derivative (e.g., \(-k\)).

Example: "The rate of growth of a population \(P\) is proportional to the square root of the population."
This translates to: \(\frac{dP}{dt} = k\sqrt{P}\), where \(k\) is a constant.

Quick Tip: If a quantity is decreasing, it is often helpful to write the equation as \(\frac{dy}{dt} = -ky\) to ensure your constant \(k\) stays positive.

2. Solving by Separating Variables

For the AQA A Level Mathematics 7357 specification, you specifically need to know how to solve first-order differential equations with separable variables. This means you can "split" the \(y\) terms and the \(x\) terms onto opposite sides of the equals sign.

The Step-by-Step Process

Don't worry if this seems tricky at first; just follow these four steps every time:

  1. Rearrange: Get everything with a \(y\) (including the \(dy\)) on one side and everything with an \(x\) (including the \(dx\)) on the other. Note: The \(dy\) and \(dx\) must always be on the top (the numerator).
  2. Integrate: Add an integral sign \(\int\) to both sides.
  3. Add the Constant: You must add a constant of integration, usually \(+ C\), on one side (typically the \(x\) side) as soon as you integrate.
  4. Solve for \(y\): Rearrange the resulting equation to make \(y\) the subject if the question asks for it.

Example: Solve \(\frac{dy}{dx} = \frac{x}{y}\)

1. Separate: \(y \ dy = x \ dx\)
2. Integrate: \(\int y \ dy = \int x \ dx\)
3. Evaluate: \(\frac{1}{2}y^2 = \frac{1}{2}x^2 + C\)

Key Takeaway: You cannot integrate until the \(dy\) and \(dx\) are separated and "upstairs." Never leave a \(dx\) in the denominator!

3. General vs. Particular Solutions

When you solve a differential equation, you will encounter two types of answers:

The General Solution

This is the solution that still contains the constant \(+ C\). It represents a whole "family" of curves that all follow the same rule.

The Particular Solution

If the question gives you boundary conditions (initial values), like "when \(x = 0, y = 5\)", you can find the specific value of \(C\). This is called the Particular Solution because it represents one specific curve from the family.

Common Mistake: Many students wait until the very end of the problem to find \(C\). It is almost always easier to plug in your \(x\) and \(y\) values immediately after you have integrated!

4. Working with Logarithms

In many AQA exam questions, you will end up with \(\frac{1}{y}\) on the left side. Integrating this gives \(\ln|y|\).

If you have \(\ln|y| = f(x) + C\), you can remove the natural log by "e-ing" both sides:
\(|y| = e^{f(x) + C}\)
Using index laws, this becomes \(y = e^{f(x)} \cdot e^C\).
Since \(e^C\) is just another constant, we often replace it with a new letter, like \(A\).
So: \(y = Ae^{f(x)}\).

Did you know? This is why exponential growth models (\(y = Ae^{kx}\)) are so common in biology—they are the direct result of the rate of growth being proportional to the size of the population!

5. Interpretation and Context

The syllabus requires you to interpret solutions in context. This means explaining what your math says about the real world.

Common Contexts:

  • Kinematics: Remember that \(v = \frac{dr}{dt}\) and \(a = \frac{dv}{dt}\). You might be asked to solve a differential equation to find displacement from acceleration.
  • Exponential Decay: Used for radioactive substances or cooling objects.
  • Limitations: You may be asked why a model might not be realistic. For example, a population model \(\frac{dP}{dt} = kP\) suggests a population will grow to infinity, which is impossible because of limited food or space.

Evaluating the Model:

When asked to "evaluate the model" or "comment on the limitations," look for what happens as time \(t\) gets very large (\(t \to \infty\)). If your equation says the temperature of a hot coffee will eventually reach \(-500\) degrees, the model is clearly limited!

Chapter Summary

  • Construct: Turn word problems into equations using \(\frac{dy}{dt}\) and \(k\).
  • Separate: Get \(y\)'s with \(dy\) and \(x\)'s with \(dx\).
  • Integrate: Use your integration techniques (Substitution, Parts, Partial Fractions).
  • Constant: Use boundary conditions to find \(C\) for a particular solution.
  • Context: Always relate your answer back to the real-world scenario and check for realistic limits.

Quick Review: To solve \(\frac{dy}{dx} = g(x)h(y)\), rearrange to \(\int \frac{1}{h(y)} \ dy = \int g(x) \ dx\). Always check if your final answer matches the units and context of the question!