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Order, Linearity, and the Language of ODEs

Every differential equation wears two labels that decide almost everything that follows: its order and whether it is linear. Learn to read those labels at a glance, and the rest of the subject's vocabulary, so a new equation tells you which toolbox to open.

Two labels on every equation

In the last guide you met the big idea: a differential equation is a rule about rates of change, and solving it means finding a whole function, not a number. Now we sort these equations into kinds. Almost everything you will do later — which method works, how many starting facts you need, whether a clean formula even exists — is decided by just two questions you can answer by looking. What is the order? And is it linear? Get fluent at reading these two labels and a strange new equation stops being intimidating; it starts announcing what it is.

Before either label, name the actors. In dy/dt = k y, the letter t is the independent variable — it just ticks forward — and y is the dependent unknown that responds to the rule. This matters more than it looks: the two labels we are about to define depend ONLY on how the unknown y and its derivatives appear. A coefficient built from t or x, however wild, never counts against you. Keep that firmly in mind and you will avoid the single most common beginner's slip.

Order: how deep into the derivatives?

The order of an equation is simply the order of the highest derivative that appears in it. In y' = k y the highest is y', so it is first order. In y'' + 3 y' + 2 y = 0 the highest is y'', so it is second order — y' and y are present too, but only the top one sets the label. A general nth-order equation reaches all the way up to y^(n). That single number is the first thing to read off, because it controls how much freedom the solutions have.

Here is why the order is worth checking first: roughly, an order-n equation carries n worth of freedom. Its general solution holds exactly n arbitrary constants, and you will need exactly n starting facts to pin down one curve. First order needs one starting value; second order needs a value and a slope — which is why Newton's law of motion, second order in time, asks for both an initial position and an initial velocity. The order tells you, before you compute anything, how many conditions the world must supply.

Linearity: the great dividing line

The second label is the more consequential one. A linear equation is one in which the unknown and all its derivatives appear only to the first power, are never multiplied by each other, and never sit inside another function like sin, exp, log, or a square root. Picture the equation as a sum of slots, one per derivative; linearity means each slot holds a coefficient times a single, bare derivative of y, and nothing more exotic. The cleanest way to recognize one is to know the standard mould it must fit.

a_n(x) y^(n) + ... + a_1(x) y' + a_0(x) y = g(x)

linear     :  y' + x^2 y = sin(x)        (y, y' first power; x-stuff is free)
nonlinear  :  y' = y^2                   (y is squared)
nonlinear  :  y' + sin(y) = 0            (y inside sine)
nonlinear  :  y y' = 1                   (y times y')
The general linear form, then four quick reads. The test always concerns y, never x.

Now the trap, worth saying twice. The coefficients a_k(x) and the right-hand side g(x) may be ANY functions of the independent variable — x^2, sin(x), e^x, even 1/x — and the equation stays linear. Linearity is a statement about the unknown alone. So y' + x^2 y = sin(x) is linear (y and y' are first power; the x-stuff is just scenery), while y' = y^2 is nonlinear because y is squared, and y' + sin(y) = 0 is nonlinear because y is buried inside the sine. Read the unknown, ignore the scenery.

Why linearity is worth obsessing over

Why does this one distinction earn so much attention? Because linear equations obey the superposition principle: in the unforced (homogeneous) case, if y1 and y2 are both solutions, then so is c1 y1 + c2 y2 for any constants. Solutions add and scale. That single fact unlocks a complete, systematic theory — we can solve essentially every linear constant-coefficient equation by a recipe, and we can build big solutions out of simple ones. The whole edifice of first-order linear integrating factors, characteristic equations, and the Laplace transform rests on this.

The rest of the working vocabulary

Within the linear family, a few more adjectives sharpen the picture, and you will hear them constantly. An equation is homogeneous when the right-hand side g(x) is zero — the equation is unforced, talking only to itself, as in a y'' + b y' + c y = 0. It is nonhomogeneous when g(x) is some nonzero driving term — a push from outside, like a y'' + b y' + c y = cos(t). That outside term has its own name, the forcing function, and the contrast between unforced and forced motion will run through the entire course.

Two more distinctions, both about the coefficients. A constant-coefficient equation has plain numbers multiplying the derivatives, like 2 y'' + 3 y' + y = 0; a variable-coefficient one lets those multipliers depend on x, like x^2 y'' + x y' + y = 0. Constant-coefficient is the friendly case we can always crack. Separately, an autonomous equation is one where the independent variable does not appear on its own — only through y — as in y' = y(1 - y); time enters only through the state, never as a clock you read directly. Autonomy is what makes the slope-field and phase-line pictures so clean later, because the rule looks the same at every instant.

Put it all together and a single equation now speaks volumes the moment you meet it. Face x^2 y'' + x y' + y = sin(x): you read second order (highest is y''), linear (y and its derivatives are all first power), variable-coefficient (the multipliers carry x), nonhomogeneous (the sin(x) on the right is a forcing term), and nonautonomous (x appears on its own). Five words, and you already know it will want two starting conditions, that superposition is available to you, and roughly which chapter holds its method. That is the payoff of the vocabulary: a label is a map.

A two-question checklist

Let us make the reading habit concrete. Given any equation, run this short checklist top to bottom; by the end you have classified it and you know what to expect. The very next guide takes the natural follow-up question — once you have a candidate formula, how do you confirm it really is a solution? — and the rest of this rung adds the initial value problem and the slope-field picture, so these labels stay with you the whole way up.

  1. Name the variables. Which letter is the independent one (usually x or t) and which is the unknown y? The two labels below judge ONLY the unknown.
  2. Find the order. Spot the highest derivative present — y', y'', up to y^(n). Its order is the equation's order, and it is the number of starting conditions you will need.
  3. Test for linearity. Do y and its derivatives appear only to the first power, never multiplied together, never inside sin/exp/log/roots? If yes, linear (superposition is yours); if you spot y^2, y y', sin(y), 1/y, it is nonlinear.
  4. If linear, add the fine print. Is the right side zero (homogeneous) or a forcing term (nonhomogeneous)? Are the coefficients constants or functions of x? Does x appear on its own (nonautonomous) or only through y (autonomous)?