What Is a First Order Differential Equation?

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The Reasoning and Methods

The order of a differential equation is determined by its highest derivative. For example,

$\frac{dy}{dx}=2xy$

is first order, whereas $\frac{d^2y}{dx^2}+y=0$ is second order.

Solving a differential equation means finding a function $y=f(x)$ that satisfies it. For the first example, separate the variables:

$\frac{1}{y}\,dy=2x\,dx$

Integrating both sides gives

$\ln|y|=x^2+C,$

so

$y=Ae^{x^2},$

where $A$ is an arbitrary constant. If an initial condition, such as $y(0)=3$, is given, then $A=3$, producing the particular solution $y=3e^{x^2}$.

To verify a proposed solution, substitute it and its derivative back into the original equation. For $y=Ae^{x^2}$, $\frac{dy}{dx}=2xAe^{x^2}=2xy$, so the equation is satisfied. The constant $A$ represents the infinitely many curves in the general solution; the initial condition selects the curve passing through $(0,3)$.

In IB Mathematics: Analysis and Approaches HL, AHL 5.18 includes these first-order methods:

For Euler’s method, start from a known point $(x_n,y_n)$ and use $x_{n+1}=x_n+h$ and $y_{n+1}=y_n+h f(x_n,y_n)$. It approximates a solution when an exact formula is unavailable or unnecessary. Its accuracy generally improves when a smaller step size is used over the same interval.

A common misconception is that the solution is a single number. The solution is normally a family of functions; an initial condition selects one particular function.

Exam Technique

In an AA HL response, identify the equation type, show the required substitution or rearrangement, integrate correctly, include the constant of integration, and apply any initial condition. Paper 1 requires analytic working without a calculator; on Papers 2 and 3, show the mathematical setup before using technology for numerical work.