3. Ordinary Differential Equations

Order of differential equations

The order of an ordinary differential equation is defined strictly as the order of the highest-order derivative appearing in the differential equation, provided the equation can be expressed as a polynomial in its derivatives.

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Definition and statement

Definition of the Order of an Ordinary Differential Equation

Let an ordinary differential equation be given in a form where it is algebraic (or can be made rational and integral) in terms of its derivatives. The order of the differential equation is the exact positive integer $n$ corresponding to the highest-order derivative $\frac{d^n y}{dx^n}$ or $y^{(n)}$ present in the relation $$F\left(x, y, \frac{dy}{dx}, \frac{d^2 y}{dx^2}, \dots, \frac{d^n y}{dx^n}\right) = 0$$.

📐 Geometric Interpretation

Geometrically, the order $n$ of a differential equation dictates the dimension of the family of curves represented by its general solution, specifically requiring $n$ independent arbitrary constants of integration. For instance, a first-order ODE represents a single-parameter family of tangent slopes, while an order $n$ ODE represents an $n$-parameter family of curves.

⚙️ Algebraic & Structural Meaning

Algebraically, determining the order requires complete freedom from radical signs and fractional powers affecting any derivative term. The expression must be cleared of radicals and rationalized with respect to all derivative operators before the order is legitimately read.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Polynomial Form in Derivatives	`Entire domain of definition`	Strictly Mandatory / Necessary	If derivatives appear inside transcendental functions like $\sin\left(\frac{dy}{dx}\right)$ or $e^{y''}$, the formal polynomial degree and sometimes even the order become undefined or infinite via series expansion.
Rationalization of Fractional Derivative Powers	`Domain where derivatives exist`	Strictly Mandatory	Failing to clear fractional exponents (e.g., $\left(\frac{d^2y}{dx^2}\right)^{1/3} = 1 + \frac{dy}{dx}$) misleads candidates into misidentifying both order and degree.
Finiteness of Derivative Order	`Discrete integer values $n \in \mathbb{N}_0$`	Essential for conclusion	Infinite series differential operators do not possess a standard finite order, invalidating standard classification.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Radical Elimination Rule

$$ \left[ \frac{d^2y}{dx^2} \right]^{p/q} = f(x, y, y') \implies \left[ \frac{d^2y}{dx^2} \right]^p = [f(x, y, y')]^q $$

Rule: Raise both sides to the power of the denominator of any fractional exponent on the highest-order derivative *before* declaring the order and degree.

Application: Use this result in the stated type of problem.

Transcendental Argument Inspection

$$ \sin\left(\frac{dy}{dx}\right) = x \implies \text{Order} = 1, \text{Degree = Not Defined} $$

Rule: If a derivative is trapped inside a non-algebraic function (sin, cos, log, exp), the order remains validly equal to the highest derivative present, but the degree is strictly undefined.

Sum of Orders in Coupled Systems

$$ \max\left( \text{Order of } x(t), \text{Order of } y(t) \right) $$

Rule: For simultaneous differential equations, the total order of the system is the order of the highest derivative present across all constituent equations.

Application: Asked in elite tier PGT mathematics papers to test depth of understanding.

Implicit Derivative Expansion

$$ \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = k \frac{d^2y}{dx^2} $$

Rule: Square both sides to eliminate the radical radicalizing the first derivative, ensuring all derivative powers are positive integers.

Direct Inspection of Constants

$$ y = c_1 e^x + c_2 e^{2x} \implies \text{Order} = 2 $$

Rule: The order of the differential equation obtained by eliminating $n$ independent arbitrary constants from a relation is always equal to $n$.

Application: Extremely high frequency in UP TGT/PGT matching equations with arbitrary constants.

Elimination of Redundant Parameters

$$ y = a \cos(x + b) \implies y'' + y = 0 \quad (\text{Order } 2) $$

Rule: Count the number of *essential* (independent) arbitrary constants. If constants can be combined (e.g., $c_1 + c_2$), simplify first before counting order.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Confusing Order with Degree

Common Pitfall: Candidates mix up the highest derivative (order) with the exponent of that highest derivative after rationalization (degree).

Counterexample: $$\left(\frac{d^3y}{dx^3}\vphantom{\frac{1}{2}}\right)^2 + \frac{dy}{dx} = 0$$

Why it fails: Order is 3 (highest derivative is third), Degree is 2 (power of the highest derivative). Candidates frequently swap these numbers.

🛡️ Exam Defense: Order = How many times you differentiate. Degree = Power of the heaviest derivative box.

⚠️ COMMON ERROR #02

Ignoring Transcendental Traps on Degree

Common Pitfall: Stating the degree of an equation where a derivative is inside a Taylor expansion or transcendental function.

Counterexample: $$\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$$

Why it fails: Order is clearly 2, but because $\sin(y')$ expands into an infinite polynomial in $y'$, the degree is not defined.

🛡️ Exam Defense: Check if any derivative argument is non-polynomial (sin, cos, log, exp). If yes, degree is undefined; order remains completely unaffected.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Order of ODE	Expressed polynomially in derivatives	Max index $n$ of $\frac{d^n y}{dx^n}$	Dimension of general solution family	Highest derivative index visible
Degree of ODE	Equation free from radical/fractional powers on derivatives	Exponent of highest derivative	Algebraic complexity of derivative term	Power of the max derivative box
General Solution	Contains $n$ independent constants for order $n$ ODE	$y = f(x, c_1, \dots, c_n)$	Full family of integral curves	Number of arbitrary parameters
Particular Solution	Arbitrary constants assigned specific values via boundary conditions	$y = f(x)$ with numerical constants	A single specific curve from the family	Initial Value Problem (IVP) given

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the order and degree of the differential equation: $$\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/2} = k \frac{d^2y}{dx^2}$$

(A) Order = 2, Degree = 1(B) Order = 2, Degree = 2(C) Order = 1, Degree = 3(D) Order = 2, Degree = 3

📚 Method A: Formal Textbook Derivation

Step 1: Square both sides to eliminate the fractional power $3/2$. We get $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = k^2 \left(\frac{d^2y}{dx^2}\right)^2$. Step 2: Identify the highest derivative, which is $\frac{d^2y}{dx^2}$ (Order = 2). Step 3: Note its exponent after squaring, which is 2 (Degree = 2).

⚡ Method B: 25-Second Exam Shortcut

Square the fractional power $3/2 \to$ power becomes 3 on LHS, power becomes 2 on RHS derivative $y''$. Thus Order = 2, Degree = 2.

💡 Tip: Do not read the degree as 3 directly from the fractional power before squaring!

CANONICAL PROBLEM 02 Dual-Method Breakdown

What is the order of the differential equation satisfied by all circles passing through the origin and having their centres on the x-axis?

(A) 1(B) 2(C) 3(D) 4

📚 Method A: Formal Textbook Derivation

Step 1: Write equation of circle passing through origin with centre $(a, 0)$: $(x-a)^2 + y^2 = a^2 \implies x^2 - 2ax + y^2 = 0$. Step 2: There is exactly 1 essential arbitrary constant ($a$). Wait, let's check: $x^2 + y^2 - 2ax = 0 \implies 2x + 2yy' - 2a = 0 \implies a = x + yy'$. Substitute $a$ back: $x^2 + y^2 - 2x(x + yy') = 0$. Step 3: This elimination involves 1 parameter, so order is 1? Let's re-verify: Equation is $(x-a)^2 + y^2 = a^2$. The center is on the x-axis, so radius is $|a|$. There is 1 parameter $a$. Hence order is 1.

⚡ Method B: 25-Second Exam Shortcut

Count essential parameters. A circle with fixed centre locus (on x-axis) has 1 parameter $a$. Order of resulting ODE is 1.

💡 Tip: Be extremely careful with geometric families. Here centre is on x-axis, so coordinates are $(a, 0)$, radius is $a$. Only one parameter $a$ exists.

Practice questions

Choose an answer to check it and see the solution:

What is the order of the differential equation $y = x \frac{dy}{dx} + \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 + b^2}$?

A 1

B 2

C 3

D Not defined

Derivation & Analysis:

Clearing the radical gives a polynomial expression in derivatives where the highest derivative is $\frac{dy}{dx}$. Thus, the order is 1.

The order of the differential equation representing the family of curves $y = c_1 e^{2x} + c_2 e^{-2x}$ is:

A 1

B 2

C 3

D 4

Derivation & Analysis:

There are 2 independent arbitrary constants ($c_1, c_2$). Eliminating them requires differentiating twice, yielding a second-order differential equation.

What is the degree of the differential equation $\frac{d^2 y}{dx^2} + \cos\left(\frac{dy}{dx}\right) = 0$?

A 1

B 2

C 0

D Not defined

Derivation & Analysis:

The derivative $\frac{dy}{dx}$ is inside a transcendental cosine function, making it impossible to express the ODE as a finite polynomial in its derivatives. Hence, degree is not defined (though order is 2).

Find the order of $x \frac{d^3y}{dx^3} + \left(\frac{d^2y}{dx^2}\right)^4 - y = 0$.

A 1

B 2

C 3

D 4

Derivation & Analysis:

The highest-order derivative present in the equation is $\frac{d^3y}{dx^3}$. Therefore, the order is 3.

The order of the differential equation of all parabolas whose axis is parallel to the y-axis is:

A 1

B 2

C 3

D 4

Derivation & Analysis:

General equation of parabola with axis parallel to y-axis is $y = ax^2 + bx + c$, which contains 3 arbitrary constants ($a, b, c$). Differentiating 3 times eliminates all constants, giving an ODE of order 3? Wait! Standard equation has 3 parameters, so order is 3.

What is the order of $\left(\frac{d^3y}{dx^3}\right)^{2/3} - 3\frac{d^2y}{dx^2} + 5\frac{dy}{dx} = 0$?

A 1

B 2

C 3

D Not defined

Derivation & Analysis:

Cube both sides to clear the fractional exponent $2/3$. The highest derivative remains $\frac{d^3y}{dx^3}$, so the order is 3.

If $y = A \sin x + B \cos x$, the corresponding differential equation is:

A $y'' + y = 0$

B $y'' - y = 0$

C $y' + y = 0$

D $y'' + yy' = 0$

Derivation & Analysis:

Differentiating twice eliminates $A$ and $B$: $y' = A\cos x - B\sin x$, $y'' = -A\sin x - B\cos x = -y \implies y'' + y = 0$ (Order 2).

The order of the differential equation of the family of circles touching the x-axis at the origin is:

A 1

B 2

C 3

D 4

Derivation & Analysis:

Equation of circles touching x-axis at origin is $x^2 + y^2 - 2ay = 0$ (1 parameter $a$). Wait, centre is $(0, a)$, radius $a$. Equation: $x^2 + (y-a)^2 = a^2 \implies x^2 + y^2 - 2ay = 0$. One parameter $a$, so order is 1? Let's check: $2x + 2yy' - 2ay' = 0 \implies a = \frac{x + yy'}{y'}$. Substituting gives an ODE of order 1.

Which of the following differential equations has order 2 and degree 1?

A $\frac{dy}{dx} + \left(\frac{d^2y}{dx^2}\right)^2 = x$

B $\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^3 = y$

C $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = 0$

D $y = x y' + (y'')^3$

Derivation & Analysis:

In option B, the highest derivative is $\frac{d^2y}{dx^2}$ with exponent 1, and no fractional powers or transcendental traps exist. Hence Order = 2, Degree = 1.

Q10

The number of arbitrary constants in the general solution of a differential equation of fourth order is:

A 2

B 3

C 4

D Infinite

Derivation & Analysis:

By definition, the general solution of an ordinary differential equation of order $n$ contains exactly $n$ independent arbitrary constants.

Common questions

No. The order of a differential equation represents the count of successive differentiations performed. It must always be a non-negative integer ($0, 1, 2, 3, \dots$). An order of 0 simply implies an algebraic equation with no derivatives.

Only fractional powers or radicals on the *highest-order derivative* dictate whether rationalization is strictly required to determine the degree. However, standard examination convention dictates clearing all radical signs across all terms before declaring the degree.

Because $\sin(y'')$ expands via Taylor series into an infinite polynomial in $y''$: $y'' - \frac{(y'')^3}{3!} + \dots = x$. Since the degree of a polynomial is the highest power of its term, an infinite series has no highest power, rendering the degree undefined.

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