Mathematics1 course

Differential Equations

Separable, linear first-order and constant-coefficient second-order equations with initial values.

About Differential Equations

A differential equation relates a quantity to its rate of change. Population growth, cooling coffee, a swinging spring and the charge in a circuit are all described this way, so the subject is where calculus turns into a modelling tool for science and engineering.

The first techniques are separation of variables and integrating factors for first-order equations. The simplest model, $\frac{dy}{dt} = ky$ , has solution $y = y_0 e^{kt}$ , which explains exponential growth and decay. Second-order linear equations with constant coefficients then describe oscillation and damping, and slope fields show the behaviour of solutions even when no formula exists.

Engineers use differential equations to design suspensions and circuits, biologists to model epidemics and populations, economists to model growth, and pharmacologists to predict how a drug's concentration falls over time. Numerical methods, which approximate solutions step by step, are how computers handle the many equations without closed-form answers.

Differential Equations

Analytic solution methods for the most common ordinary differential equations. (College)

Separable Equations

A differential equation relates a function to its derivatives. It is separable when it can be written $\frac{dy}{dx} = g(x)h(y)$ , so that all the $y$ terms can be moved to one side and all the $x$ terms to the other: $\frac{dy}{h(y)} = g(x)\,dx$ . Integrating both sides then gives the solution, with one constant of integration.

Linear First-Order Equations

A first-order linear differential equation has the form $\frac{dy}{dx} + P(x)y = Q(x)$ . Unlike separable equations, the variables cannot generally be split, but an integrating factor $\mu(x) = e^{\int P(x)\,dx}$ makes the left side the derivative of a product, $\frac{d}{dx}(\mu y)$ .

Second-Order Linear (Constant Coefficients)

A second-order linear equation with constant coefficients, $ay'' + by' + cy = 0$ , describes springs, pendulums and electrical circuits. Trying $y = e^{rx}$ turns it into the characteristic equation $ar^2 + br + c = 0$ , an ordinary quadratic whose roots determine the shape of every solution.

Full Differential Equations course outline

How to study Differential Equations

Classify the equation first (order, linear or not, separable or not); the method follows from the type.
Always apply the initial condition at the end to find the constant.
Check a solution by differentiating it and substituting back into the equation.
Sketch a slope field for simple equations to see whether solutions grow, decay or level off.

Before you start: Calculus 1 and 2: derivatives, integrals and techniques of integration.
Where it leads: Linear algebra for systems of equations, or applied courses in physics and engineering.

Key terms in Differential Equations

Order: The highest derivative in the equation. $y'' + y = 0$ is second order and needs two initial conditions to fix one solution.
Separable equation: A first-order equation that can be written $g(y)\,dy = f(x)\,dx$ and solved by integrating each side.
Integrating factor: A multiplier $e^{\int P\,dx}$ that turns $y' + Py = Q$ into an exact derivative, so both sides can be integrated.
Characteristic equation: For $ay'' + by' + cy = 0$ , the quadratic $ar^2 + br + c = 0$ whose roots give the exponential or oscillating solutions.
Equilibrium solution: A constant solution where the derivative is zero. Its stability tells whether nearby solutions move toward it or away.
Initial value problem: A differential equation together with starting values, such as $y(0) = 2$ , that picks out a single solution curve.

About Differential Equations

Differential Equations

Separable Equations

Linear First-Order Equations

Second-Order Linear (Constant Coefficients)

How to study Differential Equations

Key terms in Differential Equations

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