Differential Equations - Introduction

A population of bacteria doubles every hour. You start with 1000 bacteria at time t = 0. After one hour, there are 2000. After two hours, 4000. After three hours, 8000. The pattern is clear — but here is the deeper question: why does it double?

Because the rate at which bacteria reproduce is proportional to how many are already there. More bacteria means more reproduction, which means more bacteria, which means more reproduction. If y is the population at time t, then the rate of growth \frac{dy}{dt} is proportional to y itself:

where k is some positive constant. This single line captures the entire logic of exponential growth. It does not tell you the population at any particular time — it tells you a relationship between the population and its own rate of change.

That relationship is a differential equation. It is not an algebraic equation like x^2 - 5x + 6 = 0, where you are hunting for a number. It is an equation where the unknown is an entire function — and the equation involves not just the function, but its derivatives.

You already know how to find the answer to this one, even if you have not seen the formal method. If \frac{dy}{dt} = ky, then y must be a function whose derivative is a constant multiple of itself. The exponential function does exactly that: y = Ce^{kt}, where C is whatever the population was at time t = 0. For our bacteria, C = 1000 and k = \ln 2 (since the population doubles each hour), so y = 1000 \cdot 2^t.

That is the entire arc of this subject in miniature. You start with a relationship between a function and its derivatives. You find the function. And the answer contains a constant that gets pinned down by the specific situation.

What makes an equation "differential"

An ordinary algebraic equation involves an unknown number x and asks you to find its value. A differential equation involves an unknown function y = f(x) and asks you to find the function — and it does this by giving you a relationship between y, x, and the derivatives \frac{dy}{dx}, \frac{d^2y}{dx^2}, and so on.

Here are a few examples, each genuinely different.

Each of these involves y (the unknown function), x (the independent variable), and at least one derivative of y. That is the defining feature. No derivatives, no differential equation.

Equation (1) is the simplest kind: it says the derivative of y is 2x, so y is any antiderivative of 2x, namely y = x^2 + C. You have already been solving this type every time you computed an indefinite integral — you just didn't call it a differential equation.

Order — counting derivatives

The first thing you do when you see a differential equation is note the highest derivative that appears. That number is the order of the equation.

Equation	Highest derivative	Order
\dfrac{dy}{dx} = 2x	\dfrac{dy}{dx} (first derivative)	1
\dfrac{dy}{dx} + y = e^x	\dfrac{dy}{dx} (first derivative)	1
\dfrac{d^2y}{dx^2} + y = 0	\dfrac{d^2y}{dx^2} (second derivative)	2
\dfrac{d^3y}{dx^3} - 2\dfrac{dy}{dx} = x	\dfrac{d^3y}{dx^3} (third derivative)	3

The order tells you how many times you need to integrate (roughly speaking) to get from the equation to the solution. A first-order equation needs one integration; a second-order equation needs two. That is why second-order equations produce solutions with two arbitrary constants, third-order with three, and so on.

For the JEE and most of class 12, the focus is on first-order ordinary differential equations — and even within that, on specific types that have neat solution methods. The remaining articles in this series cover those types one by one.

Degree — the power of the highest derivative

After order, the second classification is degree. Write the equation so that the highest-order derivative is free of radicals and fractions (as a polynomial in that derivative). Then the degree is the power (exponent) to which the highest-order derivative is raised.

Take the equation

The highest derivative is \frac{d^2y}{dx^2}, so the order is 2. It appears raised to the power 3, so the degree is 3.

Now take

This involves a square root. To find the degree, you must first clear the radical — square both sides:

Now the highest derivative \frac{d^2y}{dx^2} appears with power 2. So the order is 2 and the degree is 2.

There is an important subtlety: the degree is only defined when the equation can be written as a polynomial in its derivatives. If you have something like

there is no way to write this as a polynomial in \frac{dy}{dx} — the sine function prevents it. For such equations, the degree is not defined.

Where differential equations come from — formation

There is a beautiful reverse question: instead of solving a differential equation to get a family of curves, can you start with a family of curves and find the differential equation they all satisfy?

Take the family of all parabolas with vertex at the origin, opening upward:

Here a is an arbitrary constant — different values of a give different parabolas, all sharing the same vertex. This is a one-parameter family of curves.

To find the differential equation, eliminate the arbitrary constant a.

Differentiate y = ax^2 with respect to x:

From the original equation, a = \frac{y}{x^2}. Substitute:

So the differential equation is

or equivalently, x\,\frac{dy}{dx} = 2y. Notice: the constant a has vanished. The differential equation captures what all the parabolas in this family have in common — at every point on any such parabola, the slope \frac{dy}{dx} equals \frac{2y}{x}.

The general principle:

• A family with one arbitrary constant produces a first-order DE (differentiate once, eliminate one constant).
• A family with two arbitrary constants produces a second-order DE (differentiate twice, eliminate two constants).
• A family with n arbitrary constants produces an n-th order DE.

Here is a second example. Take the family of all circles centred at the origin:

One arbitrary constant (r), so expect a first-order DE. Differentiate implicitly:

Simplify:

The constant r is already gone — it disappeared the moment you differentiated, because the derivative of r^2 (a constant) is zero. The resulting DE says: at every point on any circle centred at the origin, the quantity x + y\frac{dy}{dx} is zero. Geometrically, this says the radius to any point is perpendicular to the tangent there — which is a theorem you already know from coordinate geometry.

Types of solutions

When you solve a differential equation, the answer is not a single function — it is a family of functions. Understanding the different types of solutions is essential.

General solution

The general solution of a differential equation of order n is the solution containing n arbitrary constants. It represents the entire family of curves that satisfy the equation.

For instance, the general solution of \frac{dy}{dx} = 2x is

where C is an arbitrary constant. For each value of C, you get a different parabola — all of them have the same shape, just shifted vertically.

Particular solution

A particular solution is what you get when you assign a specific value to every arbitrary constant. It is one specific curve from the general family.

If someone tells you that y = 3 when x = 1, you can pin down C: 3 = 1^2 + C, so C = 2. The particular solution is y = x^2 + 2. That's one specific parabola from the family.

The information "y = 3 when x = 1" is called an initial condition (or a boundary condition, depending on context). For a first-order equation, one initial condition pins down one constant; for a second-order equation, you need two conditions.

Singular solution

Some differential equations have solutions that cannot be obtained from the general solution by any choice of the constant. These are called singular solutions. They arise when the process of eliminating the constant introduces extra solutions.

A classic example: the equation

has the general solution y = (x - C)^2 (a family of parabolas shifted horizontally). But y = 0 is also a solution — the zero function satisfies the equation, since both sides become zero — and it cannot be recovered from (x - C)^2 for any finite value of C. The line y = 0 is a singular solution: it is the envelope of the family of parabolas.

Singular solutions are important but relatively rare in the first course. You should know they exist, but most of the equations you solve in the following articles will have only the general and particular solutions.

Worked examples

Common confusions

A few points that consistently trip students up.

• "Order and degree are the same thing." They are not. Order is which derivative is highest; degree is what power that derivative is raised to. The equation \left(\frac{dy}{dx}\right)^5 = x has order 1 and degree 5. The equation \frac{d^5y}{dx^5} = x has order 5 and degree 1.

• "Every differential equation has a degree." Only equations that can be written as polynomials in their derivatives have a well-defined degree. If the equation involves e^{y'} or \sin(y''), the degree is not defined.

• "The general solution of a first-order DE has no constant." The opposite — a first-order DE's general solution has exactly one arbitrary constant. A second-order DE's general solution has two. The number of constants equals the order.

• "A particular solution is less correct than the general solution." Both are equally valid. The general solution represents every curve in the family. A particular solution is one specific curve that satisfies both the DE and the given conditions. In applications, the particular solution is usually what you want — the general solution is a stepping stone.

• "Formation of a DE means you can always eliminate constants easily." Sometimes the elimination is algebraically messy. The strategy is always the same — differentiate enough times and use the equations to eliminate the constants — but the algebra can require care, especially with trigonometric or exponential families.

Going deeper

Why arbitrary constants appear

The appearance of arbitrary constants in the solution is not an accident — it is structurally inevitable. Here is why.

Solving a differential equation of order n involves, at bottom, performing n integrations. Each integration introduces one constant of integration. So the general solution of an n-th order ODE contains exactly n arbitrary constants. This is a theorem, not just a pattern — under standard smoothness conditions, the solution space of an n-th order linear ODE is an n-dimensional vector space, and the n constants parametrise that space.

Existence and uniqueness

A natural question: does every differential equation actually have a solution? And if it does, is the solution unique once you specify initial conditions?

The answer to both is "yes, under mild conditions." The Picard-Lindelöf theorem guarantees that if the right-hand side of \frac{dy}{dx} = f(x, y) is continuous and satisfies a Lipschitz condition in y, then for any initial condition y(x_0) = y_0, there is exactly one solution in some neighbourhood of x_0.

You will not need to verify these conditions for any exam problem — the equations you meet will always have well-behaved right-hand sides. But the theorem is worth knowing because it tells you that the process of "solving" a differential equation is not fishing in the dark — there really is a unique answer waiting, and the methods you learn in the next articles will find it.

Linear vs nonlinear

A differential equation is linear if the unknown function y and all its derivatives appear only to the first power and are not multiplied together. So y'' + 3y' - 2y = \sin x is linear, but y \cdot y' = x is not (because y and y' are multiplied), and (y')^2 = y is not (because y' is squared).

Linear equations are far easier to solve systematically. They have a rich theory — superposition of solutions, variation of parameters, characteristic equations. Most of the first course on differential equations (and most of this article series) focuses on first-order equations, some linear and some not, each with its own solution technique.

Where this leads next

You now know what a differential equation is, how to classify it, and what its solutions look like. The next articles teach you how to actually solve specific types.

• First Order — Variable Separable — the simplest first-order equations, where you can separate x and y onto different sides and integrate directly.
• First Order — Reducible to Separable — equations that don't look separable at first but become so after a substitution, including homogeneous equations.
• First Order — Linear DE — equations of the form \frac{dy}{dx} + Py = Q, solved using integrating factors.
• First Order — Reducible to Linear — Bernoulli equations and other forms that a substitution turns into linear equations.
• First Order — Exact Equations — equations where the left-hand side is already the total differential of some function.