AHL 5.18—1st order DE’s – Euler method, variables separable, integrating factor, homogeneous DE using sub y=vx

A first-order differential equation (DE) relates a function $y(x)$ to its first derivative $\frac{dy}{dx}$. In this subtopic, you will master four key techniques for solving or approximating solutions to first-order DEs:

1. Euler's Method , numerical approximation
2. Variables Separable , algebraic separation and integration
3. Integrating Factor , for linear first-order DEs
4. Homogeneous Substitution ($y=vx$) , for homogeneous DEs

First-Order Differential Equation: An equation involving an unknown function $y(x)$ and its first derivative $\frac{dy}{dx}$, written in the general form $\frac{dy}{dx}=f(x,y)$.

Initial Condition: A known value $y(x_0)=y_0$ that specifies a particular solution from the family of general solutions of a DE.

Choosing the right method depends on the structure of the DE , learning to recognise each type quickly is a key exam skill.

The Euler method is a numerical technique for approximating solutions to first-order DEs when an exact analytical solution is difficult or impossible to find. The idea is to take small steps along the solution curve, using the derivative at each point to estimate the next.

The Core Idea:
Given a point $(x_n,y_n)$ on the solution curve, the tangent line at that point is:
$y=y_n+f(x_n,y_n)(x-x_n)$
where $f(x_n,y_n)=\frac{dy}{dx}$ evaluated at $(x_n,y_n)$.

For a small step size $h$, we assume the solution curve stays close to this tangent line, so the next point $(x_{n+1},y_{n+1})$ is approximated by:

$\boxed{x_{n+1}=x_n+h}$
$\boxed{y_{n+1}=y_n+h\cdot f(x_n,y_n)}$

This generates a sequence of $(x_n,y_n)$ values that approximately follow the solution curve.

Step Size (h): The fixed increment added to $x$ at each iteration of Euler's method. A smaller $h$ gives greater accuracy but requires more computational steps.