Difficulty: Intermediate | Prerequisites: Calculus I & II (integration techniques, basic derivatives), familiarity with functions of one and several variables.
Differential equations sit at the heart of advanced calculus and applied mathematics. They describe how quantities change in relation to one another, which makes them the natural language for modelling anything that evolves: motion, heat flow, population growth, chemical reactions.
If you have a solid grip on integration and differentiation, you are ready for this material. If not, revisit those fundamentals first, because every solution method here leans on them. This topic bridges pure calculus and the applied sciences, so expect to see it referenced repeatedly in physics, engineering, and chemistry courses.
Differential equations relate a function to its own derivatives. They come in several flavours (ordinary vs. partial, linear vs. nonlinear), and the method you choose to solve one depends on what type you are dealing with. Mastering the classifications and the core solution techniques is the foundation for nearly every applied maths or science course that follows.
Differential equation
An equation involving a function and one or more of its derivatives. In simple terms, it tells you how a quantity's rate of change depends on the quantity itself (or other variables).
Ordinary differential equation (ODE)
A differential equation with only one independent variable (often time, t) and one or more dependent variables. Think of it as: "the function depends on just one input."
Partial differential equation (PDE)
A differential equation with multiple independent variables (for example, both position x and time t). In simple terms, the function depends on more than one input, so you take partial derivatives rather than ordinary ones.
Linear differential equation
A differential equation in which the dependent variable and its derivatives appear only to the first power, with no products between them. Think of it as: "no squaring, no multiplying y by dy/dx, nothing nonlinear."
Nonlinear differential equation
A differential equation where the dependent variable or its derivatives appear in nonlinear combinations (squared terms, products of y and its derivatives, etc.). These are typically harder to solve and may not have closed-form solutions.
Separation of variables
A technique for ODEs in which you rearrange the equation so that all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other, then integrate both sides.
Integrating factor
A function you multiply through a linear ODE to convert the left-hand side into an exact derivative, making the equation directly integrable. In simple terms, it is a clever multiplier that tidies the equation into something you can integrate in one step.
Power series solution (series solution)
A method where you assume the solution takes the form of an infinite power series, substitute it into the ODE, and solve for the coefficients term by term.
Numerical methods
Techniques that approximate the solution of a differential equation when an exact (analytical) answer is impractical or impossible. Common examples include Euler's method, Runge-Kutta methods, and finite element methods.
ODE vs. PDE is the first fork in the road. Count the independent variables: one gives you an ODE, more than one gives you a PDE.
Linear vs. nonlinear is the second. Check whether the dependent variable and its derivatives appear only in first-power, additive combinations. If so, the equation is linear; otherwise, nonlinear.
Classification matters because it determines which solution methods apply. Choosing the wrong method for the wrong type wastes time and produces errors.
Separation of variables
Works when you can cleanly isolate all y terms (including dy) on one side and all x terms (including dx) on the other.
Once separated, integrate both sides independently.
This is often the first method to try for a first-order ODE.
Integrating factors
Used for first-order linear ODEs of the form dy/dx + P(x)y = Q(x).
The integrating factor is μ(x) = e^(∫P(x) dx).
Multiply through, recognise the left side as d/dx [μ(x) · y], then integrate.
Substitution
Introduce a new variable to simplify the equation (e.g., v = y/x for homogeneous ODEs).
Solve the simpler equation in the new variable, then substitute back.
Series solutions
Assume y = Σ aₙxⁿ, plug into the ODE, and match coefficients of like powers of x.
Particularly useful near ordinary points where other methods stall.
Numerical methods
Euler's method: step-by-step linear approximation using the slope at each point. Simple but can accumulate error quickly.
Runge-Kutta methods: more sophisticated, sampling the slope at multiple points per step for greater accuracy.
Finite element methods: break the domain into small elements and solve locally, then stitch together. Common in engineering simulations.
General first-order linear ODE: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^(∫P(x) dx)
Separable form: g(y) dy = f(x) dx → ∫g(y) dy = ∫f(x) dx + C
Power series assumption: y = Σ (n=0 to ∞) aₙxⁿ
Physics: Newton's second law (F = ma) is an ODE. Maxwell's equations (electromagnetism) and the Schrödinger equation (quantum mechanics) are PDEs. If you continue into any branch of physics, differential equations are the toolkit.
Chemistry: Reaction rate laws are expressed as ODEs. The rate at which a reactant is consumed or a product is formed over time is modelled by differential equations, which is how chemists predict concentrations at any future moment.
Engineering: Structural analysis, fluid dynamics, circuit design, and control systems all rely on differential equations. The finite element method, for instance, is how engineers simulate stress on a bridge before building it.
Students often assume that "ODE" and "linear" mean the same thing. They do not. An ODE can be nonlinear (e.g., dy/dx = y²), and a PDE can be linear.
Separation of variables does not work on every ODE. If you cannot cleanly isolate y terms from x terms, you need a different method. Forcing a separation that is not there leads to algebraic errors.
The integrating factor method applies specifically to linear first-order ODEs. Applying it to a nonlinear equation will not produce a valid solution.
Numerical methods are approximations, not exact solutions. A smaller step size improves accuracy but increases computation. Students sometimes treat numerical output as exact, which can be misleading on exam questions that ask for analytical solutions.
⚠️ Classification questions are common: expect to be given an equation and asked whether it is an ODE or PDE, linear or nonlinear, and what order it is. These are quick marks if you know the definitions.
⚠️ The integrating factor formula (μ = e^(∫P(x) dx)) is heavily tested. Know how to identify P(x) from the standard form and compute μ without hesitation.
⚠️ Separation of variables is typically the first solution method tested. Practice recognising separable equations quickly.
⚠️ Be prepared to justify why a particular method applies to a given equation type. Exam questions often penalise using the wrong method even if you reach a plausible-looking answer.
True or false: Every ODE is linear.
Fill in the blank: The integrating factor for dy/dx + P(x)y = Q(x) is μ(x) = ________.
True or false: Separation of variables requires you to get all y terms on one side and all x terms on the other before integrating.
Fill in the blank: A differential equation with two or more independent variables is called a ________ differential equation.
True or false: Euler's method gives an exact solution to an ODE.
Answers: 1. False. 2. e^(∫P(x) dx). 3. True. 4. Partial. 5. False (it is an approximation).
Q: What distinguishes an ODE from a PDE?
A: An ODE involves only one independent variable; a PDE involves two or more independent variables.
Q: Given dy/dx + 3y = 6, identify P(x), Q(x), and the integrating factor.
A: P(x) = 3, Q(x) = 6, and the integrating factor is μ(x) = e^(∫3 dx) = e^(3x).
Q: Is the equation dy/dx = y² linear or nonlinear? Explain.
A: Nonlinear, because the dependent variable y appears squared. A linear equation requires y and its derivatives to appear only to the first power.
Q: Name two numerical methods for approximating solutions to differential equations.
A: Euler's method and Runge-Kutta methods (finite element methods is also acceptable).
Q: When would you choose a series solution over separation of variables?
A: When the ODE cannot be separated into isolated y and x terms, or when the equation has variable coefficients that make other analytical methods impractical. Series solutions are especially useful near ordinary points.
This material connects directly to integration techniques from Calculus II, since every solution method ultimately involves integration. It also feeds forward into courses on mathematical physics, control theory, and numerical analysis, where these equations are solved in progressively more complex settings.
If you are taking linear algebra concurrently, note that systems of linear ODEs can be solved using matrix methods (eigenvalues and eigenvectors), which bridges these two courses neatly.
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