3. Ordinary Differential Equations

Degree of differential equations

The degree of a differential equation is defined as the highest power of its highest-order derivative, provided the equation can be expressed as a polynomial equation in its derivatives.

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Definition and statement

Definition of Degree of an Ordinary Differential Equation

Let an ordinary differential equation be expressed in a form that is polynomial in its derivatives, given by $$P_n \left(\frac{d^n y}{dx^n}\right)^n + \dots + P_1 \frac{dy}{dx} + P_0 y = Q(x)$$, where each $P_i$ and $Q$ are functions of $x$ (or constants). The degree of the differential equation is the highest exponent (positive integer) of the highest-order derivative present in the equation after it has been cleared of any fractional or radical exponents affecting the derivative terms.

📐 Geometric Interpretation

Geometrically, the order of a differential equation corresponds to the dimension of the family of curves (number of arbitrary constants in the primitive), whereas the degree reflects the algebraic multiplicity or algebraic complexity of the slope function defining the tangent vector field at any point $(x, y)$.

⚙️ Algebraic & Structural Meaning

Algebraically, the differential equation must be a polynomial expression in the derivative operators $D^k y = \frac{d^k y}{dx^k}$. If transcendental functions like $\sin\left(\frac{dy}{dx}\right)$ or $\exp\left(\frac{d^2y}{dx^2}\right)$ contain derivatives as arguments, the degree is undefined because they cannot be expanded into finite polynomial forms.

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 (e.g., $\cos(y') = x$), the degree is undefined as it lacks a polynomial expansion.
Rationalization of Fractional Exponents	`Domain where derivatives exist`	Strictly Mandatory	Failure to clear fractional powers (e.g., $(y'')^{1/3} = y' + 5$) leads to incorrect identification of degree. Both sides must be raised to the power required to eliminate fractions.
Finiteness of Order and Degree	`Real domain`	Essential for conclusion	Infinite series involving derivatives do not possess a well-defined finite degree.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Radical Clearance Rule

$$ \left( \frac{d^2 y}{dx^2} \right)^{p/q} = f(x, y, y') \implies \text{Degree is the power corresponding to } p \text{ after rationalizing } q. $$

Rule: Raise both sides of the equation to the power of the denominator $q$ of any fractional exponent of the highest-order derivative before reading the degree.

Application: Directly tested in UP PGT where equations like $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = k \frac{d^2y}{dx^2}$ are given.

Transcendental Trap Elimination

$$ \sin\left(\frac{dy}{dx}\right) = x \implies \text{Degree = Undefined} $$

Rule: If a derivative appears as the argument of $\sin$, $\cos$, $\log$, or $e^x$, check if it can be expanded into an infinite series. If so, degree is not defined.

Polynomial in Derivatives Check

$$ a_n(x) (y^{(n)})^m + \dots + a_0(x)y = 0 \implies \text{Order} = n, \text{Degree} = m $$

Rule: Once cleared of radicals and fractions, the exponent of the highest-order derivative is the degree.

Application: Standard 1-mark question in UP TGT Mathematics.

Inverse Trigonometric / Logarithmic Clearing

$$ \tan^{-1}\left(\frac{dy}{dx}\right) = x \implies \frac{dy}{dx} = \tan x \implies \text{Order} = 1, \text{Degree} = 1 $$

Rule: Apply inverse operations (like taking $\tan$ on both sides) to isolate the derivative into a polynomial form before determining order and degree.

Product & Quotient Simplification

$$ y'' = \sqrt{1 + (y')^2} \implies (y'')^2 = 1 + (y')^2 \implies \text{Degree} = 2 $$

Rule: Square or expand algebraic radical expressions involving derivatives completely before stating the degree.

Rapid Elimination Trick

$$ \text{Options with 'Not Defined' are correct for } \sin(y'), e^{y''}, \text{ or } \log(y''') $$

Rule: Instantly look for transcendental functions wrapping derivatives; if present, select 'Not Defined' without solving further.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

The Fractional Power Superficial Read

Common Pitfall: Candidates read the exponent of the highest-order derivative without clearing fractional exponents first.

Counterexample: $(\frac{d^2y}{dx^2})^{1/2} + \frac{dy}{dx} = 0$

Why it fails: Looking at the power $1/2$ might lead someone to guess degree $1/2$ or $1$ improperly. Squaring gives $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$, yielding degree $1$.

🛡️ Exam Defense: Always eliminate fractional indices on derivatives by raising both sides to the LCM of denominators.

⚠️ COMMON ERROR #02

Confusing Order and Degree

Common Pitfall: Swapping the definitions of order and degree under exam stress.

Counterexample: $5 \frac{d^3y}{dx^3} + 4\left(\frac{dy}{dx}\right)^7 = x^2$

Why it fails: Order is 3 (highest derivative), degree is 1 (power of highest derivative, NOT 7). Candidates often mistakenly report degree 7.

🛡️ Exam Defense: Order is the maximum number of differentiations; degree is the exponent of that specific highest-order derivative term.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Order of DE	Existence of derivatives up to order n	Max derivative index $n$	Dimension of family of curves	Highest derivative present
Degree of DE	Polynomial form in derivatives	Highest exponent of max derivative	Algebraic multiplicity of slope field	Power of highest derivative after radical clearance
Linear DE	Degree 1 in y and all derivatives	$a_n(x)y^{(n)} + \dots + a_0(x)y = Q(x)$	Superposition principle holds	No products of y and its derivatives
Non-Linear DE	Fails linearity conditions	Degree > 1 or products of $y, y'$ present	Non-linear tangent vector fields	Products or powers of $y$ and derivatives

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

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

(A) 1(B) 2(C) 3(D) Not defined

📚 Method A: Formal Textbook Derivation

Step 1: Identify highest-order derivative: $\frac{d^2y}{dx^2}$ (order 2). Step 2: Clear fractional power $3/2$ by squaring both sides: $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^3 = k^2 \left(\frac{d^2y}{dx^2}\right)^2$. Step 3: Observe the exponent of $\frac{d^2y}{dx^2}$ is 2. Hence, degree = 2.

⚡ Method B: 25-Second Exam Shortcut

Square both sides to eliminate the $3/2$ power on the left, making the power of $y''$ equal to 2 directly.

💡 Tip: Very frequent in UP PGT. Remember that radius of curvature formula gives degree 2.

CANONICAL PROBLEM 02 Dual-Method Breakdown

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

(A) 1(B) 2(C) 0(D) Not defined

📚 Method A: Formal Textbook Derivation

Step 1: Check if the equation is a polynomial in its derivatives. Step 2: The term $\cos\left(\frac{dy}{dx}\right)$ can be expanded as an infinite series $1 - \frac{(y')^2}{2!} + \dots$, which contains infinite powers of the derivative. Step 3: Therefore, it cannot be expressed as a polynomial equation in derivatives. Degree is not defined.

⚡ Method B: 25-Second Exam Shortcut

Spot $\cos$, $\sin$, $\log$, or $e$ containing a derivative -> immediately mark 'Not defined'.

💡 Tip: Do not fall for the apparent power 1 of $y''$ in front.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: Find the degree of the differential equation $\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + y^2 = 0$.

A 1

B 2

C 3

D Not defined

Derivation & Analysis:

The highest-order derivative is $\frac{d^2 y}{dx^2}$ (order 2), and its exponent (power) is 1. Thus, the degree is 1.

MCQ 2: What is the degree of $\left(\frac{d^3y}{dx^3}\right)^2 + 4\frac{d^2y}{dx^2} - 5\frac{dy}{dx} = e^x$?

A 3

B 2

C 1

D Not defined

Derivation & Analysis:

Highest-order derivative is $\frac{d^3y}{dx^3}$ with order 3. Its power in the given polynomial form is 2. Hence, degree = 2.

MCQ 3: Find the degree of $\sqrt{\frac{dy}{dx}} - \frac{dx}{dy} = 2$.

A 1

B 2

C 1/2

D Not defined

Derivation & Analysis:

Rewrite as $(y')^{1/2} - \frac{1}{y'} = 2$. Multiply by $y'$: $y' - 1 = 2y' \implies (y')^2 - 2y'(1) - 1 = 0$ after rationalizing. The highest power of $y'$ becomes 2.

MCQ 4: The degree of the differential equation $\frac{dy}{dx} + \tan(y) = x$ is:

A 0

B 1

C 2

D Not defined

Derivation & Analysis:

Here $\tan(y)$ contains $y$, not $\frac{dy}{dx}$. The equation is linear in $\frac{dy}{dx}$ with power 1. Thus, order = 1, degree = 1.

MCQ 5: The degree of the differential equation $y = x\frac{dy}{dx} + \sqrt{a^2 \left(\frac{dy}{dx}\right)^2 + b^2}$ is:

A 1

B 2

C 4

D Not defined

Derivation & Analysis:

Isolate the radical: $y - x y' = \sqrt{a^2 (y')^2 + b^2}$. Square both sides: $(y - x y')^2 = a^2 (y')^2 + b^2$. Expanding yields terms in $(y')^2$, so the degree is 2.

MCQ 6: Find the degree of $\ln\left(\frac{dy}{dx}\right) = ax + by$.

A 1

B e

C 0

D Not defined

Derivation & Analysis:

Exponentiating gives $\frac{dy}{dx} = e^{ax+by}$. Wait, here the derivative is inside the logarithm! $\ln(y') = ax+by \implies y' = e^{ax+by}$. Since $y'$ is NOT inside the log (the log is applied TO $y'$), let's check: $\frac{dy}{dx} = e^{ax+by}$ is polynomial in $y'$, so degree is 1. (Caution: If it were $\log(y'') + \dots$, evaluate carefully. Here $y' = e^{ax}e^{by}$, degree is 1.)

MCQ 7: What is the degree of the differential equation corresponding to all circles passing through the origin?

A 1

B 2

C 3

D 4

Derivation & Analysis:

Family of circles equation has 2 arbitrary constants, so order is 2. The resulting differential equation is of second order and first degree (degree = 1).

MCQ 8: The degree of $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{3/4} = \left(\frac{d^2y}{dx^2}\right)^{1/2}$ is:

A 3

B 2

C 4

D 1

Derivation & Analysis:

Raise both sides to the power 4 (LCM of denominators 4 and 2): $\left(1 + (y')^2\right)^3 = \left((y'')^{1/2}\right)^4 = (y'')^2$. The highest derivative is $y''$ and its power is 2. Thus, degree = 2.

MCQ 9: If $y = x \frac{dy}{dx} + \frac{a}{\frac{dy}{dx}}$, its degree is:

A 1

B 2

C 3

D Not defined

Derivation & Analysis:

Multiply by $\frac{dy}{dx}$: $y \frac{dy}{dx} = x \left(\frac{dy}{dx}\right)^2 + a$. The highest derivative is $\frac{dy}{dx}$ and its maximum power is 2? Wait, highest power across all terms is 2, but degree is the power of the HIGHEST-ORDER derivative. Here order is 1, highest-order derivative is $\frac{dy}{dx}$, and its highest power is 2. Thus, degree = 2.

Q10

MCQ 10: The degree of the differential equation $\frac{d^3y}{dx^3} + 2\left(\frac{d^2y}{dx^2}\right)^2 - \frac{dy}{dx} = \sin x$ is:

A 3

B 2

C 1

D 6

Derivation & Analysis:

Highest-order derivative is $\frac{d^3y}{dx^3}$ with order 3. Its exponent in the given equation is 1. Therefore, degree = 1.

Common questions

No. By definition, the degree must be a positive integer ($n \in \mathbb{N}$). Fractional or negative powers must always be rationalized and cleared before determining the degree.

If derivatives appear as arguments of transcendental functions (e.g., $\sin(y'), e^{y''}, \ln(y'')$), the degree is officially 'not defined'.

Multiplying by polynomial expressions in $x$ or $y$ does not change the degree. However, multiplying by fractional powers or derivatives can alter the apparent polynomial form, which is why complete rationalization is mandatory first.

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