Solving & Interpreting Differential Equations (AQA A-Level Mathematics): Revision Notes

8.3.5 Solving & Interpreting Differential Equations

Solving and interpreting differential equations is a crucial aspect of A Level Maths, particularly in modelling real-world scenarios. Let's break down the process into clear steps, covering both the mathematical techniques for solving differential equations and how to interpret the solutions in context.

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1. Solving Differential Equations

There are several types of differential equations, and each type has specific methods for finding the solution. Below are the common types you might encounter:

a) First-Order Differential Equations

These involve the first derivative of the unknown function.

i. Separable Differential Equations:

When a differential equation can be written in the form:

$\frac{dy}{dx} = g(x)h(y)$

Steps to Solve:

• Separate the Variables: Move all terms involving $y$ to one side and all terms involving $x$ to the other.
• Integrate Both Sides: Integrate with respect to $x$ and $y$.
• Solve for $y$ : If possible, solve explicitly for $y$ .

ii. Linear First-Order Differential Equations:

These are of the form:

$\frac{dy}{dx} + P(x)y = Q(x)$

Steps to Solve:

• Find the Integrating Factor (IF):

$IF = e^{\int P(x) \, dx}$

• Multiply the whole equation by the IF to make the left side an exact derivative.
• Integrate the resulting equation.
• Solve for $y$.

b) Second-Order Differential Equations

These involve the second derivative of the unknown function and often appear in problems related to motion and oscillations.

i. Homogeneous Equations:

These are of the form:

$\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0$

Steps to Solve:

• Find the characteristic equation: This is a quadratic equation obtained by substituting $y = e^{mx}$ .
• Solve the characteristic equation: The nature of the roots (real and distinct, real and repeated, or complex) determines the form of the solution.
• Form the general solution using the roots of the characteristic equation.

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2. Interpreting Solutions

After solving a differential equation, it's important to interpret the solution in the context of the problem. Here's how you might approach it:

a) Initial Conditions

• Often, the problem will provide initial conditions, such as $y(0) = y_0$ , to find a particular solution.
• Substitute the initial conditions into the general solution to find the specific values of the constants $C_1$ and $C_2$.

b) Physical Interpretation

• The solution should be related back to the physical scenario described by the problem.
• For example, in a population growth model where $y(t) = P_0e^{kt}$ , $y(t)$ might represent the population at time $t$ , with $P_0$ being the initial population and $k$ the growth rate.

c) Long-Term Behaviour

• Consider what happens to $y(x)$ as $x$ becomes very large or very small.
• This can give insight into the stability of the system or the long-term predictions of the model.

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Summary

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