Find the unknown function and its independent variables
Classifying is not about putting on labels. It tells you which tools are worth trying. The first question is always: what is unknown, and what does it change with?
Classification is a language for describing structure. It narrows down the methods worth trying, but a label never guarantees that you can compute an answer. On this page x and y are dimensionless mathematical variables, and y′ means dy/dx.
In y′=y, the unknown is a function y(x). Although both x and y appear, there is only one independent variable, x: y is the dependent variable, set by x. So this is an ordinary differential equation (ODE).
This equation uses partial derivatives: differentiate in t with x held fixed, and in x with t held fixed. It is a partial differential equation (PDE). This series is mainly about ODEs and does not go into how to solve PDEs here.
Order: how high do the derivatives go?
The order is how many layers of differentiation there are. Only the highest layer counts. How long the formula is, or how complicated its coefficients are, makes no difference.
y′ is the first derivative and y″ the second. The order is the highest derivative the equation actually contains. Cancel any terms that cancel identically first, then look at what is left; don’t count prime marks or terms.
So “higher order” and “higher degree” are not the same thing. Higher order adds layers of derivatives; higher degree is about the power a derivative is raised to. Don’t read a page on first-order equations of higher degree as if it were about second-order equations.
Linearity: does the unknown appear only once?
Linearity is about how the unknown function shows up: on its own, to the first power. It has nothing to do with whether a graph is a straight line.
On an interval you are studying, a linear ODE can be written as
y and its derivatives appear only to the first power, never multiplied together and never inside a square, sin, an exponential or another nonlinear function. The coefficients may depend only on the independent variable; they can’t quietly contain the unknown y.
Read an equation that looks long
(1+x²)y″ + sin(x)y′ + y = eˣThe unknowns are y, y′ and y″. Their coefficients, 1+x², sin(x) and 1, depend only on x.
Find the highest order and check linearity
Highest derivative y″; coefficient 1+x² > 0It stays second order on the whole real line, and it is linear. sin(x) is not sin(y), so it does not break linearity.
See whether there is a forcing term
g(x) = eˣ ≠ 0In standard form, the right side contains no y and is not zero, so the equation is linear and nonhomogeneous. A nonhomogeneous equation can still be linear.
Short, yet nonlinear
y′=y²: the unknown is squared.
y y′=x: the unknown is multiplied by its own derivative.
y′+sin(y)=0: the unknown sits inside sin.
Long, yet linear
(1+x²)y″+sin(x)y′+y=eˣ: variable coefficients and a forcing term do not make it nonlinear. Being linear does not promise a simple elementary formula either.
Only when g(x) is zero everywhere is the equation linear homogeneous. So y′=y, which is the same as y′−y=0, is linear homogeneous, while y′−y=x is linear nonhomogeneous. These linear labels do not apply to nonlinear equations.
Dividing y′=y² by y to get y′/y=y does not make it linear: the coefficient 1/y on the left still depends on the unknown, and the division also throws away the zero solution.
Degree: first check for a polynomial in the derivatives
The degree is the power the highest derivative is raised to. A first-order equation can have a high degree, and a high-order equation can be first degree. Keep the two ideas apart.
This page uses the common definition: once the equation is written as a polynomial in its derivatives, the highest power of the highest-order derivative is the degree of the equation. The coefficients may still contain x and y. Check that this definition applies before you write down a number.
| Equation | Order | Degree | Why |
|---|---|---|---|
| y′=y² | 1 | 1 | The highest derivative, y′, is not squared. |
| (y′)²=y | 1 | 2 | The highest derivative is squared. |
| y″+(y′)³=0 | 2 | 1 | Look at the power of y″, not the largest power anywhere (3). |
| sin(y′)=y | 1 | undefined | Not a polynomial in the derivative. |
Squaring, taking roots and dividing are not free tidying up. They can add or remove solutions, and they can change how the degree appears. For example, y′=y and (y′−y)²=0 have the same real solutions, but written as polynomials they have different powers. Degree describes one specific form of an equation; on its own it is not a permanent label for the solutions or the difficulty.
If you square to remove a root, keep the original sign condition and check the result in the original equation. If you solve sin(y′)=y with arcsin, the principal value misses the other angle branches. This page does not force a degree onto an equation through steps that change it.
Four questions for any equation
Ask four questions in order, and an unfamiliar equation falls into place: how many independent variables, what order, is it linear, what degree.
In order: How many independent variables does the unknown function depend on? How high do the derivatives go? Does the unknown appear linearly? If the equation is a polynomial in its derivatives, what power is the highest derivative raised to?
1. Is y″+(y′)³=0 second order and third degree?
See the full reasoning
No. It is a second-order, first-degree, nonlinear ODE. The highest derivative is y″, raised to the power 1. (y′)³ makes it nonlinear, but it does not make the degree 3.
2. In x y′+y=1, the coefficient is zero at x=0. Can you just divide by x?
See the full reasoning
Only on intervals where x≠0. At x=0 the original equation requires y(0)=1, while after dividing, y′+y/x=1/x is undefined there. You can still recognize the equation as linear, but to use the theory of normalized first-order equations you have to work on separate intervals.
3. Is y′−y=sin(x) nonlinear?
See the full reasoning
It is a first-order, first-degree, linear nonhomogeneous ODE. The input to sin is the known x; nothing nonlinear is done to the unknown y. The right side, sin(x), is not zero everywhere, so the equation is not linear homogeneous.
Classification describes what an equation looks like. “Existence and Uniqueness” asks something more basic: given a starting point, is there a solution, and is there only one?
Sources and scope
Definitions and the conditions of theorems were checked against the authors and university materials below. The explanations, worked examples and interactive plots here are written independently. Decimals on the plots are rounded displays; the mathematical conclusions rest on derivation and substitution. This page does not prove the general existence and uniqueness theorems it cites.
- Jiří Lebl · Classification: basic definitions of ODE and PDE, order, linearity and homogeneity.
- NCERT / IIT Kanpur SATHEE · Differential Equations: order and degree for equations that are polynomials in their derivatives.