Types of Equations

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What you are solving for, and how the unknown appears

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An equation says two expressions have the same value. You often solve for an unknown number, unknown function, or unknown vector.

Two labels help before the list:

• Conditional equation — true only for some values. Example: $2x + 3 = 7$ (solution: $x = 2$).
• Identity — true for every allowed value. Example: $(x + 1)^2 = x^2 + 2x + 1$.

The table below is a family map, not a strict taxonomy. One expression can fit more than one row (a quadratic can also be transcendental if it uses $\sin x$).

Family	You usually find	One example
Algebraic	Numbers in powers and roots	$x^2 - 4 = 0$
Differential	Functions and their rates of change	$\frac{dy}{dx} = ky$
Transcendental	Values inside exp, log, trig	$2^x = 8$
Functional	A whole function $f$	$f(x+y) = f(x) + f(y)$
Integral	A function under an integral	$f(x) = g(x) + \int_a^b K(x,t)f(t),dt$
Parametric	A curve via a parameter $t$	$x = \cos t$, $y = \sin t$
Matrix / vector	A vector $x$ in $Ax = b$	$2x_1 + x_2 = 5$ and $x_1 + 3x_2 = 7$ (same system as §7)

For how symbols are read in any expression, see The Grammar of Mathematics: Operators, Functions, Relations, and Symbols Explained. For the number sets those symbols use, see Kinds of Numbers: A Map from Natural to Complex.

1. Algebraic equations

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Built from add, subtract, multiply, divide, powers, and roots on variables and constants.

• Linear — highest power of the variable is $1$. Graph: a line. Example: $2x + 3 = 7$.
• Quadratic — highest power is $2$. Graph: a parabola. Example: $x^2 - 5x + 6 = 0$.
• Polynomial — sum of terms with non-negative integer powers (cubic, quartic, …). Example: $x^3 - 4x + 1 = 0$.
• Rational — polynomials in numerators or denominators. Example: $\frac{1}{x} + \frac{2}{x-1} = 3$.
• Radical — variable inside a root. Example: $\sqrt{x + 2} = 5$.

Systems: several equations at once, e.g. $x + y = 5$ and $2x - y = 1$ (two lines meeting at one point).

2. Differential equations

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The unknown is a function, and the equation uses derivatives (rates of change). Common in physics, biology, and control.

• ODE (ordinary) — one independent variable. Example: $\frac{dy}{dx} + 2y = 0$.
• PDE (partial) — several variables and partial derivatives. Example (wave equation): $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$.

3. Transcendental equations

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Uses functions that are not just polynomials and roots: exponential, logarithm, trigonometry.

• Exponential / logarithmic: $2^x = 8$ or $\ln x = 5$.
• Trigonometric: $\sin x + \cos x = 1$ on a chosen interval.

Some transcendental equations still have a closed-form solution: you write the unknown as one exact formula using finitely many steps with arithmetic, powers, roots, exp, log, and trig. From $\ln x = 5$ you get $x = e^5$; from $2^x = 8$ you get $x = 3$.

Others have no elementary closed form — no single short formula with only those operations. Example: find $x$ such that $e^x = x$. Then you use a numerical method: a fixed recipe that approximates the value to as many decimals as you need. Textbooks call that a numerical solution or approximation; they do not use the phrase “open form” as the opposite of closed form for these cases.

4. Functional equations

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The unknown is a whole function $f$: a rule that sends each input $x$ to an output $f(x)$. You want the formula for that rule, not a single number.

Same idea as data work: you may have inputs and measured outputs and ask “what function produced these?” Fitting a line in regression is one simple case. Here the twist is different: a functional equation gives a law that $f$ must obey for every allowed input at once — not only at the rows in a table.

Concrete $f$: if $f(x) = 2x$, then $f(3) = 6$ and $f(-1) = -2$. The equation below must stay true when you plug in any $x$ (or any pair $x, y$), not just one test value.

One function, one variable — find all $f$ that work:

• $f(x + 1) = f(x) + 2$ for every real $x$. Try $f(x) = 2x + 1$: left $2(x+1)+1 = 2x+3$, right $(2x+1)+2 = 2x+3$. Many solutions share the same slope; the puzzle is to list or describe them all.

Two functions, one variable — find pairs $(f, g)$ that work:

• $f(g(x)) = x$ for every $x$ in the domain. Then $g$ undoes $f$ (inverse functions). Example: $f(x) = 2x$, $g(x) = x/2$, because $f(g(x)) = 2 \cdot (x/2) = x$.

One function, two inputs (still one rule $f$):

• $f(x + y) = f(x) + f(y)$ for every $x, y$. Add the inputs, add the outputs. Example that works: $f(x) = 3x$, because $3(x+y) = 3x + 3y$. The question is which other formulas for $f$ still satisfy the rule; if $f$ is also continuous, the answer is exactly multiply by a constant: $f(x) = cx$ (such as $c = 3$).

5. Integral equations

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The unknown function appears under an integral sign. Many models in scattering and some machine-learning kernels use this form.

Example: $f(x) = g(x) + \int_a^b K(x, t) f(t) dt$, where $K(x,t)$ is the kernel (a known function that weights $f(t)$ at each $t$) and $g(x)$ is known.

Differential and integral equations are close cousins; many ODEs can be rewritten as integrals.

6. Parametric equations

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Instead of $y$ as a function of $x$ alone, both coordinates follow a parameter $t$ (often time).

Example: $x = \cos t$, $y = \sin t$ traces a circle as $t$ runs from $0$ to $2\pi$.

7. Matrix and vector equations

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One compact statement for many scalar equations at once — the language of linear algebra and large systems.

Example: $Ax = b$ with matrix $A$, unknown vector $x$, and known vector $b$. The same two lines as in the table, stacked:

\[\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \end{bmatrix}.\]

For a deeper pass on matrices, see Matrices: A Complete Revision Guide.

Quick checks

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1. Is $\sqrt{x} = 4$ algebraic? Yes — radical algebraic.
2. Is $\frac{dy}{dt} = -y$ algebraic? No — differential (ODE).
3. Is $e^x = 10$ polynomial? No — transcendental.
4. Are $x = \cos t$, $y = \sin t$ one equation? Two equations sharing parameter $t$ — a parametric system.

Related reading

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• The Grammar of Mathematics: Operators, Functions, Relations, and Symbols Explained
• Kinds of Numbers: A Map from Natural to Complex
• Matrices: A Complete Revision Guide
• Start Here — Data Science & AI on dasarpAI

Hashtags

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