Basic Concepts — Order and Degree

What Is a Differential Equation?

Ordinary equations like $x^2 - 3x + 3 = 0$ or $x + y = 7$ involve only variables. An equation like

$$x\frac{dy}{dx} + y = 0$$

involves a derivative of the dependent variable — and that changes everything. In general:

Definition: An equation involving derivatives of the dependent variable with respect to the independent variable is called a differential equation.

Such equations arise wherever change is modelled — physics, chemistry, biology, economics — because nature usually specifies rates, and solving the differential equation recovers the quantity itself.

Ordinary versus partial

1. A differential equation involving derivatives with respect to only one independent variable is an ordinary differential equation (ODE) — e.g. $2\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 = 0$.
2. Equations with derivatives with respect to several independent variables are partial differential equations — beyond this course. From here, "differential equation" means an ordinary one.

Notation

$$\frac{dy}{dx} = y', \qquad \frac{d^2y}{dx^2} = y'', \qquad \frac{d^3y}{dx^3} = y''', \qquad \frac{d^ny}{dx^n} = y_n$$

The dash notation keeps long equations readable; $y_n$ avoids a forest of dashes for higher orders.

Key Point: A differential equation is a relationship between a function and its rates of change. "Solving" it will mean finding every function $y(x)$ that makes the relationship true — a function-valued answer, not a number-valued one.

Order and Degree

Order

Definition: The order of a differential equation is the order of the highest order derivative of the dependent variable appearing in it.

1. $\dfrac{dy}{dx} = e^x$ — highest derivative $\frac{dy}{dx}$: order 1.
2. $\dfrac{d^2y}{dx^2} + y = 0$ — highest derivative $\frac{d^2y}{dx^2}$: order 2.
3. $\dfrac{d^3y}{dx^3} + x^2\left(\dfrac{d^2y}{dx^2}\right)^3 = 0$ — highest derivative $\frac{d^3y}{dx^3}$: order 3 (the power on $y''$ is irrelevant to order).

Degree — check the polynomial condition first

Definition: When the equation is a polynomial equation in its derivatives $y', y'', y''', \ldots$, the degree is the highest power (a positive integer) of the highest order derivative.

The three NCERT touchstones:

1. $y''' + 2\left(y''\right)^2 - y' + y = 0$ — polynomial in $y''', y'', y'$: degree $= 1$ (power of $y'''$).
2. $\left(y'\right)^2 + y' - \sin^2 y = 0$ — polynomial in $y'$ (the $\sin^2 y$ involves $y$, not a derivative — harmless): degree $= 2$.
3. $y' + \sin\left(y'\right) = 0$ — $\sin$ of a derivative: not a polynomial in $y'$, so the degree is not defined.

The rules in brief

1. Order is always defined; degree only when the polynomial condition holds.
2. Order and degree (when defined) are always positive integers.
3. Functions of $x$ or of $y$ alone ($e^x$, $\sin y$, $\log x$) never spoil the degree — only functions of derivatives ($\sin y'$, $e^{y'}$, $\cos y''$) do.

Key Point: Two separate questions, two separate scans. First scan for the highest derivative (order — always answerable). Then ask "polynomial in the derivatives?" — if yes, read off the power of the highest derivative; if no, write "degree not defined", which is itself a full-credit answer.

[JEE Tip] JEE and boards both bait with equations needing clearing first: $y'' = \sqrt{1 + y'}$ must be squared to $\left(y''\right)^2 = 1 + y'$ before reading degree $2$. Clear roots and fractions of derivatives first, then apply the definitions — but never try to "clear" a $\sin(y')$; that stays undefined.

Solved Examples

Example 1: A clean first-order equation

Find the order and degree, if defined, of $\dfrac{dy}{dx} - \cos x = 0$.

Solution:

1. Order: highest derivative is $\frac{dy}{dx}$ — order $1$.
2. Degree: polynomial in $y'$ ($\cos x$ involves only $x$); power of $y'$ is $1$.

Final Answer: order $1$, degree $1$.

Example 2: Products of derivatives

Find the order and degree, if defined, of $xy\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 - y\dfrac{dy}{dx} = 0$.

Solution:

1. Order: highest derivative is $\frac{d^2y}{dx^2}$ — order $2$.
2. Degree: the equation is polynomial in $y''$ and $y'$; the highest power of $y''$ is $1$ (the square sits on $y'$, which doesn't count).

Final Answer: order $2$, degree $1$.

Example 3: A derivative inside an exponential

Find the order and degree, if defined, of $y''' + y^2 + e^{y'} = 0$.

Solution:

1. Order: highest derivative $y'''$ — order $3$.
2. Degree: $e^{y'}$ is an exponential of a derivative — the equation is not polynomial in its derivatives.

Final Answer: order $3$, degree not defined. (The $y^2$ term is irrelevant — powers of $y$ itself never matter.)

Example 4: Sine of a third derivative

Find the order and degree, if defined, of $\dfrac{d^4y}{dx^4} + \sin\left(y'''\right) = 0$.

Solution:

1. Order: highest derivative $\frac{d^4y}{dx^4}$ — order $4$.
2. Degree: $\sin(y''')$ breaks the polynomial condition.

Final Answer: order $4$, degree not defined.

Example 5: Watch which derivative carries the power

Find the order and degree, if defined, of $\left(\dfrac{ds}{dt}\right)^4 + 3s\,\dfrac{d^2s}{dt^2} = 0$.

Solution:

1. Order: highest derivative is $\frac{d^2s}{dt^2}$ — order $2$.
2. Degree: polynomial in both derivatives; the power of the highest derivative $\frac{d^2s}{dt^2}$ is $1$ — the fourth power on the lower-order $\frac{ds}{dt}$ does not set the degree.

Final Answer: order $2$, degree $1$.

Takeaway: Degree reads the exponent on the highest-order derivative only. A big power on a lower derivative is a decoy — the single most common order/degree error.

Example 6: Both order and degree above 1

Find the order and degree, if defined, of $\left(y'''\right)^2 + \left(y''\right)^3 + \left(y'\right)^4 + y^5 = 0$.

Solution:

1. Order: highest derivative $y'''$ — order $3$.
2. Degree: polynomial in all derivatives; power of $y'''$ is $2$.

Final Answer: order $3$, degree $2$.