3. Ordinary Differential Equations

Formation of differential equations

The formation of a differential equation involves eliminating arbitrary constants from a given relation representing a family of curves, resulting in an equation whose order equals the number of independent constants.

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Definition and statement

Principle of Elimination of Arbitrary Constants

Let $f(x, y, c_1, c_2, \dots, c_n) = 0$ be an equation representing a family of curves containing $n$ independent arbitrary constants. By differentiating the relation $n$ times successively with respect to $x$ and eliminating the $n$ parameters $c_1, c_2, \dots, c_n$ among the $n+1$ resulting equations, we obtain an ordinary differential equation of order $n$: $$F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \dots, \frac{d^n y}{dx^n}\right) = 0$$

📐 Geometric Interpretation

Geometrically, an equation with $n$ parameters defines an $n$-parameter family of curves (e.g., all circles of fixed radius, or all parabolas with a common vertex). Differentiating and eliminating constants constructs a differential equation whose slope field or curvature at every point is satisfied by every member of the geometric family.

⚙️ Algebraic & Structural Meaning

Algebraically, the number of independent arbitrary constants dictates the exact order of the resulting differential equation. Eliminating constants reduces the transcendence or parameter dependence of the function down to a differential operator relation.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Independence of Constants	`Domain of definition $D \subset \mathbb{R}$`	Strictly Mandatory / Necessary	If constants are not independent (e.g., $c_1 + c_2$ grouped as a single constant), overcounting the parameters leads to an over-ordered differential equation.
Sufficient Differentiability	`Open interval $(a, b)$`	Strictly Mandatory	Failure of the function to possess continuous derivatives up to order $n$ prevents legitimate elimination and substitution.
Algebraic Elimination Validity	`Real coordinate space`	Essential for conclusion	Division by zero or improper algebraic manipulation during elimination introduces extraneous solutions or singular solutions.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Single Parameter Family (Lines/Circles)

$$y = mx + c \implies \frac{d^2y}{dx^2} = 0$$

Rule: For a family of straight lines involving two parameters ($m$ and $c$), the differential equation is always of order 2: $d^2y/dx^2 = 0$.

Application: undefined

Circle Family with Center on Axis

$$(x-h)^2 + y^2 = r^2 \implies \text{Order } 2$$

Rule: Count the number of essential independent constants. If center $(h,k)$ is on the x-axis, $k=0$, leaving only $h$ and $r$ (2 constants), yielding a 2nd order DE.

Exponential Growth/Decay Family

$$y = Ae^{mx} + Be^{nx} \implies \frac{d^2y}{dx^2} - (m+n)\frac{dy}{dx} + mn y = 0$$

Rule: For linear combinations of exponentials, the differential equation corresponds directly to the auxiliary equation roots $m$ and $n$.

Product Form / Logarithmic Separation

$$y = c e^{f(x)} \implies \frac{1}{y}\frac{dy}{dx} = f'(x)$$

Rule: When a single constant is multiplicative with an exponential, take the natural logarithm or differentiate once to immediately eliminate $c$ without simultaneous equations.

Application: Saves 45 seconds on exponential curve problems.

Trigonometric Phase Families

$$y = A \cos(x + \theta) \implies \frac{d^2y}{dx^2} + y = 0$$

Rule: Two parameters ($A$ and $\theta$) in a simple harmonic motion equation always yield a 2nd order differential equation free of parameters.

Rapid Option Elimination via Order Check

$$n = \text{Number of essential constants}$$

Rule: Always count the exact number of independent arbitrary constants first. Immediately eliminate any option whose order does not match $n$.

Application: Crucial time management strategy for 125 questions in UP PGT.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

Redundant or Combined Constants

Common Pitfall: Counting $c_1 + c_2$ or $c_1 c_2$ as two independent constants instead of simplifying them into a single parameter before counting.

Counterexample: $y = c_1 e^{x + c_2} = (c_1 e^{c_2})e^x = c e^x$

Why it fails: Though written with two constants, it simplifies to only ONE essential constant. Treating it as 2nd order results in an incorrect differential equation.

🛡️ Exam Defense: Always simplify algebraic expressions and combine constants to find the true number of independent parameters before differentiating.

⚠️ COMMON ERROR #02

Mistaking Degree for Order

Common Pitfall: Confusing the number of arbitrary constants with the degree of the differential equation rather than its order.

Counterexample: $y = cx + c^2$

Why it fails: Has 1 constant ($c$), so order is 1, but eliminating $c$ produces a quadratic term in $dy/dx$, making its degree 2.

🛡️ Exam Defense: Number of constants = Order of DE. Degree is the highest power of the highest derivative after rationalization.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
1-Parameter Family	1 independent constant $c$	$\frac{dy}{dx} = f(x, y)$	Family of curves intersecting at specific slopes	Single constant present in equation
n-Parameter Family	$n$ independent constants	$\frac{d^n y}{dx^n} = \Phi(x, y, \dots)$	Higher-order geometric curve manifold	Multiple constants like $c_1, c_2, \dots, c_n$
Trigonometric Family	Amplitude and phase angle	$\frac{d^2y}{dx^2} \pm k^2 y = 0$	Simple harmonic oscillations	Sine and cosine terms with constants
Exponential Family	Growth/decay coefficients	$\frac{dy}{dx} - ky = 0$	Exponential scaling curves	Base $e$ raised to linear functions of $x$

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Find the differential equation of all circles passing through the origin and whose centers lie on the x-axis.

(A) $(x^2 - y^2)\frac{dy}{dx} = 2xy$(B) $(x^2 + y^2)\frac{dy}{dx} = 2xy$(C) $(x^2 - y^2)\frac{dy}{dx} = xy$(D) $(x^2 + y^2) = 2xy \frac{dy}{dx}

📚 Method A: Formal Textbook Derivation

Step 1: Equation of circle with center $(a, 0)$ and radius $a$ passing through origin is $(x-a)^2 + y^2 = a^2 \implies x^2 - 2ax + y^2 = 0$. Step 2: Differentiate with respect to $x$: $2x - 2a + 2y\frac{dy}{dx} = 0 \implies a = x + y\frac{dy}{dx}$. Step 3: Substitute $a$ back into the original equation to eliminate $a$. Step 4: Simplify to get $(x^2 - y^2)\frac{dy}{dx} = 2xy$.

⚡ Method B: 25-Second Exam Shortcut

Observe that parameter $a$ appears linearly. Isolate $2a = \frac{x^2+y^2}{x}$ and substitute derivative directly.

💡 Tip: Standard UP PGT repeat question. Memorize the resulting differential equation.

CANONICAL PROBLEM 02 Dual-Method Breakdown

The differential equation corresponding to $y = c_1 e^{2x} + c_2 e^{-3x}$ is:

(A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0$(B) $\frac{d^2y}{dx^2} - \frac{dy}{dx} - 6y = 0$(C) $\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0$(D) $\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + 6y = 0$

📚 Method A: Formal Textbook Derivation

Step 1: Here roots of auxiliary equation are $m_1 = 2$ and $m_2 = -3$. Step 2: Sum of roots $= 2 + (-3) = -1$; Product of roots $= (2)(-3) = -6$. Step 3: Form equation $\frac{d^2y}{dx^2} - (\text{sum})\frac{dy}{dx} + (\text{prod})y = 0 \implies \frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0$.

⚡ Method B: 25-Second Exam Shortcut

Use the quadratic operator relation $(D - m_1)(D - m_2)y = 0$ where $D = \frac{d}{dx}$.

💡 Tip: Direct quadratic auxiliary formula saves 1 minute.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The order of the differential equation of all conics centered at the origin is:

A 1

B 2

C 3

D 4

Derivation & Analysis:

General equation of conics centered at origin involves quadratic forms with coefficients that reduce to 2 independent parameters after normalization.

MCQ 2: What is the differential equation representing the family of curves $y = a \sin(x + b)$?

A $\frac{d^2y}{dx^2} + y = 0$

B $\frac{d^2y}{dx^2} - y = 0$

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

D $\left(\frac{dy}{dx}\right)^2 + y^2 = a^2$

Derivation & Analysis:

Two arbitrary constants ($a$ and $b$). Differentiating twice gives $\frac{d^2y}{dx^2} = -a\sin(x+b) = -y \implies \frac{d^2y}{dx^2} + y = 0$.

MCQ 3: The differential equation of the family of lines passing through the origin is:

A $x \frac{dy}{dx} - y = 0$

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

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

D $\frac{d^2y}{dx^2} = 0$

Derivation & Analysis:

Equation is $y = mx$. Differentiating gives $\frac{dy}{dx} = m$. Eliminating $m$ gives $y = x\frac{dy}{dx} \implies x\frac{dy}{dx} - y = 0$.

MCQ 4: The number of arbitrary constants in the general solution of a differential equation of order $n$ is:

A n - 1

B n

C n + 1

D Independent of n

Derivation & Analysis:

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

MCQ 5: The differential equation formed by eliminating $a$ and $b$ from $y = e^x(a \cos x + b \sin x)$ is:

A $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0$

B $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 2y = 0$

C $\frac{d^2y}{dx^2} - y = 0$

D $\frac{d^2y}{dx^2} + y = 0$

Derivation & Analysis:

Using operator methods or double differentiation for $e^{ax}(c_1\cos bx + c_2\sin bx)$, the auxiliary equation has roots $1 \pm i$, leading to $\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 2y = 0$.

MCQ 6: The differential equation of the family of parabolas with vertex at $(0,0)$ and axis along the y-axis is:

A $x \frac{dy}{dx} - 2y = 0$

B $2x \frac{dy}{dx} - y = 0$

C $x \frac{dy}{dx} - y = 0$

D $\frac{dy}{dx} - 2xy = 0$

Derivation & Analysis:

Equation is $x^2 = 4ay$. Differentiating once: $2x = 4a \frac{dy}{dx}$. Dividing gives $\frac{x^2}{2x} = \frac{4ay}{4a\frac{dy}{dx}} \implies \frac{x}{2} = \frac{y}{dy/dx} \implies x\frac{dy}{dx} - 2y = 0$.

MCQ 7: If $y = c_1 e^{3x} + c_2 e^{-3x}$, the corresponding differential equation is:

A $\frac{d^2y}{dx^2} - 9y = 0$

B $\frac{d^2y}{dx^2} + 9y = 0$

C $\frac{d^2y}{dx^2} - 3y = 0$

D $\frac{d^2y}{dx^2} + 3y = 0$

Derivation & Analysis:

Roots are $3$ and $-3$. Sum = $0$, Product = $-9$. Thus $\frac{d^2y}{dx^2} - 9y = 0$.

MCQ 8: The differential equation of all non-vertical lines in the Cartesian plane is of order:

A 1

B 2

C 3

D Infinite

Derivation & Analysis:

Non-vertical lines are given by $y = mx + c$, which has 2 independent constants ($m$ and $c$), hence order 2.

MCQ 9: Eliminate $c$ from $xy = c^2$ to form the differential equation:

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

B $x \frac{dy}{dx} - y = 0$

C $x \frac{dy}{dx} + 2y = 0$

D $\frac{dy}{dx} + xy = 0$

Derivation & Analysis:

Differentiating $xy = c^2$ with respect to $x$: $x\frac{dy}{dx} + y = 0$, since derivative of $c^2$ is 0.

Q10

MCQ 10: The differential equation of the family of circles touching the y-axis at the origin is:

A $(x^2 - y^2)\frac{dy}{dx} - 2xy = 0$

B $(x^2 + y^2)\frac{dy}{dx} - 2xy = 0$

C $(x^2 - y^2) + 2xy\frac{dy}{dx} = 0$

D $x^2 + y^2 - 2xy\frac{dy}{dx} = 0$

Derivation & Analysis:

Center is $(a, 0)$ and passes through origin with radius $a$, equation is $x^2 + y^2 - 2ax = 0$. Differentiating and eliminating $a$ yields $(x^2 - y^2)\frac{dy}{dx} - 2xy = 0$.

Common questions

The order of the differential equation is strictly equal to the number of independent arbitrary constants present in the given relation of the family of curves.

Failing to simplify algebraic expressions. For example, $y = c_1 \sin(x + c_2)$ appears to have two constants, but expanding it gives $c_1 \cos c_2 \sin x + c_1 \sin c_2 \cos x$, which collapses into two linear coefficients $A\sin x + B\cos x$, properly counting as 2 constants.

Yes! Substitute the given family equation or its derivatives into the options, or check if differentiating the option yields an identity consistent with the original curve family.

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