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Differential Equations — Order, Degree & Formation

Master Order, Degree, and Formation of Differential Equations for JEE. Learn degree rules, eliminating constants, Clairaut's equation, and key curve families. Includes fully verified JEE practice questions.

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A Differential Equation (DE) is an equation relating a function with its derivatives. Almost every physical law — Newton's law of cooling, radioactive decay, population growth, electrical circuits — is expressed as a DE. For JEE, differential equations carry significant weight across all three paper types: formation of DEs (eliminating arbitrary constants), identifying order and degree, and solving them. This first topic covers the foundational classification and the critical skill of forming a DE from its general solution — a skill that is tested directly in JEE Main and Advanced every year. Mastering formation is the key that unlocks the solving topics that follow.

1. Order and Degree of a Differential Equation

Order: The order of the highest derivative present in the DE.

Degree: The power (exponent) of the highest-order derivative, after the DE is written as a polynomial in derivatives (i.e., after clearing all roots and fractions involving derivatives).

Differential Equation	Order	Degree	Reason
$\frac{d y}{d x} + y = x^{2}$	1	1	Highest derivative is dy/dx, raised to power 1
$\frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x})}^{3} = x$	2	1	Highest derivative is y'', power 1; (y')³ is lower order
${(\frac{d^{2} y}{d x^{2}})}^{2} + \frac{d y}{d x} = 0$	2	2	Highest derivative y'' is raised to power 2
${[1 + {(\frac{d y}{d x})}^{2}]}^{3 / 2} = \frac{d^{2} y}{d x^{2}}$	2	2	Cube both sides: [1+(y')²]³ = (y'')² → highest derivative y'' raised to 2
$\frac{d^{3} y}{d x^{3}} + y \frac{d y}{d x} = \sin x$	3	1	Highest derivative y''' raised to 1
$y = x \frac{d y}{d x} + \sqrt{1 + {(\frac{d y}{d x})}^{2}}$	1	2	Rearrange: y−xy' = √(1+(y')²); square: (y−xy')² = 1+(y')² → degree 2

⚠️ Degree is UNDEFINED when…

The DE involves transcendental functions of derivatives — e.g., sin(dy/dx), e^(dy/dx), log(y''). Such DEs cannot be written as polynomials in the derivative, so the degree is not defined.

Example: $\sin (\frac{d y}{d x}) + y = x ⟹$ degree not defined (but order = 1).

2. General Solution vs Particular Solution

General Solution: Contains as many arbitrary constants as the order of the DE. It represents a family of curves.
Particular Solution: Obtained by assigning specific values to arbitrary constants using initial conditions (ICs). It represents a single curve from the family.
Singular Solution: A solution not obtainable from the general solution for any value of C (not tested in JEE Main, rare in Advanced).

An nth-order DE → general solution has n arbitrary constants → need n ICs for a particular solution.

3. Formation of a Differential Equation

Strategy: If the general solution has n arbitrary constants, differentiate n times, then eliminate all n constants to get the DE.

Example 1 — One Constant: $y = A e^{2 x}$

Differentiate once: $\frac{d y}{d x} = 2 A e^{2 x} = 2 y$

DE: $\frac{d y}{d x} - 2 y = 0$

Example 2 — Two Constants: $y = A \sin x + B \cos x$

$y^{'} = A \cos x - B \sin x$

$y^{″} = - A \sin x - B \cos x = - y$

DE: $y^{″} + y = 0$

Example 3 — Two Constants: y = Ax² + Bx

y' = 2Ax + B; y'' = 2A, so A = y''/2

From y': B = y' − 2Ax = y' − xy''

Substituting into y: y = (y''/2)x² + (y'−xy'')x = x²y''/2 + xy' − x²y''

Simplify: x²y'' − 2xy' + 2y = 0

DE: x²y'' − 2xy' + 2y = 0 (verified: substituting y = Ax²+Bx gives 0 ✓)

Example 4 (JEE Level) — Family of circles: x² + y² = r²

Differentiate: 2x + 2y·(dy/dx) = 0 → x + y·(dy/dx) = 0

DE: x + y·dy/dx = 0 or x dx + y dy = 0

Example 5 (JEE Advanced Level) — y² = 4a(x − a)

This involves one parameter a. Differentiate: 2y·y' = 4a → a = yy'/2

Substitute back: y² = 4·(yy'/2)·(x − yy'/2) = 2yy'(x − yy'/2)

y² = 2yy'x − y²(y')² → y = 2y'x − y(y')²

DE: y(y')² − 2xy' + y = 0 (Clairaut-type structure)

4. Number of Arbitrary Constants = Order of DE

Family of curves	No. of constants	Order of DE
y = Ae^x (1 constant)	1	1
y = A sin x + B cos x (2 constants)	2	2
y = Ae^x + Be^(2x) + Ce^(3x) (3 constants)	3	3
y = Ax + B/x (2 constants)	2	2

Formation Strategy — Systematic Elimination of Constants

Step	Action
1	Count the arbitrary constants (= n). This is the order of the DE.
2	Differentiate the general solution n times to get n equations.
3	Eliminate all n constants using these n+1 equations (original + n derivatives).
4	The remaining equation — involving only x, y, y', y'', … — is the required DE.

JEE Power Tips — Order, Degree & Formation

Degree requires the DE to be polynomial in derivatives first. Always clear radicals and fractional powers of derivatives before reading the degree. For [1+(y')²]^(3/2) = y'', square both sides → [1+(y')²]³ = (y'')² → degree = 2.
Degree of a DE involving sin, cos, e, log of a derivative is NOT DEFINED. Degree requires an algebraic (polynomial) relationship. sin(y') = x has undefined degree.
In formation: the order equals the number of arbitrary constants you need to eliminate. If the family has 2 constants → differentiate twice → 3 equations → eliminate both constants → order-2 DE.
JEE ADV Clairaut's equation: y = px + f(p) where p = dy/dx. Differentiating: p = p + x·(dp/dx) + f'(p)·(dp/dx) → (dp/dx)[x + f'(p)] = 0. Either dp/dx = 0 (p = C, general solution y = Cx + f(C)) or x + f'(p) = 0 (parametric singular solution).
A family of parabolas y² = 4a(x−a) leads to a DE with only one constant despite two letters — a appears twice but as one parameter. Differentiate once to get the DE.

Practice Questions

Q1 (JEE Main): The order and degree of the differential equation ${(\frac{d^{2} y}{d x^{2}})}^{2} + \cos (\frac{d y}{d x}) = 0$ are respectively:

A) (2, 2)
B) (2, undefined)
C) (1, 2)
D) (2, 1)

Answer: B) (2, undefined).

Explanation:
Order: The highest order derivative present in the differential equation is the second derivative $\frac{d^{2} y}{d x^{2}}$ . Thus, the order is 2.
Degree: The degree is the highest power of the highest order derivative, provided the differential equation is a polynomial equation in its derivatives. Because this equation contains the term $\cos (\frac{d y}{d x})$ (a transcendental function of the first derivative), it cannot be expressed as a polynomial in the derivatives. Therefore, the degree is strictly undefined.

Q2 (JEE Main): Form the differential equation for the family of circles passing through the origin with their centres on the x-axis: $x^{2} + y^{2} = 2 a x$ .

Explanation:

Given the family of curves:
$x^{2} + y^{2} = 2 a x --- (Equation 1)$

Differentiating both sides with respect to $x$ :
$2 x + 2 y \frac{d y}{d x} = 2 a$

Divide by 2 to isolate the arbitrary constant $a$ :
$a = x + y \frac{d y}{d x}$

Substitute this value of $a$ back into Equation 1 to eliminate the constant:
$x^{2} + y^{2} = 2 x (x + y \frac{d y}{d x})$
$x^{2} + y^{2} = 2 x^{2} + 2 x y \frac{d y}{d x}$

Rearranging the terms:
$y^{2} - x^{2} = 2 x y \frac{d y}{d x}$

Differential Equation: $2 x y \frac{d y}{d x} = y^{2} - x^{2}$

Q3 (JEE Main): Find the differential equation of the family of curves $y = A e^{x} + B e^{- x}$ .

Explanation:

There are two arbitrary constants ( $A$ and $B$ ), so we need to differentiate twice.
Given: $y = A e^{x} + B e^{- x}$

Differentiate once with respect to $x$ :
$y^{'} = A e^{x} - B e^{- x}$

Differentiate a second time with respect to $x$ :
$y^{''} = A e^{x} + B e^{- x}$

Notice that the right side of the second derivative is exactly our original function $y$ .
$y^{''} = y$

Differential Equation: $y^{''} - y = 0$

Q4 (JEE Advanced level): The degree of the differential equation $y = x \frac{d y}{d x} + \sqrt{a^{2} {(\frac{d y}{d x})}^{2} + b^{2}}$ is:

Explanation:

To find the degree, we must first free the differential equation from fractional powers (radicals) regarding its derivatives. Let $p = \frac{d y}{d x}$ .

Rearrange to isolate the radical:
$y - x p = \sqrt{a^{2} p^{2} + b^{2}}$

Square both sides to eliminate the square root:
$(y - x p)^{2} = a^{2} p^{2} + b^{2}$
$y^{2} - 2 x y p + x^{2} p^{2} = a^{2} p^{2} + b^{2}$

This equation is now a proper polynomial in terms of $p$ (i.e., $\frac{d y}{d x}$ ). The highest power of $p$ is 2 (from the $x^{2} p^{2}$ and $a^{2} p^{2}$ terms).

Degree = 2. (The Order is 1, since only the first derivative appears).

Q5 (Board / JEE Main): Form the differential equation by eliminating $A$ and $B$ from the function: $y = A \cos (2 x) + B \sin (2 x)$ .

Explanation:

There are two arbitrary constants, so we differentiate twice.
Differentiating once with respect to $x$ (applying the chain rule):
$y^{'} = - 2 A \sin (2 x) + 2 B \cos (2 x)$

Differentiating a second time:
$y^{''} = - 4 A \cos (2 x) - 4 B \sin (2 x)$

Factor out $- 4$ from the expression:
$y^{''} = - 4 (A \cos (2 x) + B \sin (2 x))$

The expression inside the parentheses is our original function $y$ :
$y^{''} = - 4 y$

Differential Equation: $y^{''} + 4 y = 0$

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Differential Equations — Order, Degree & Formation

1. Order and Degree of a Differential Equation

2. General Solution vs Particular Solution

3. Formation of a Differential Equation

Example 1 — One Constant: $y = A e^{2 x}$

Example 2 — Two Constants: $y = A \sin x + B \cos x$

Example 3 — Two Constants: y = Ax² + Bx

Example 4 (JEE Level) — Family of circles: x² + y² = r²

Example 5 (JEE Advanced Level) — y² = 4a(x − a)

4. Number of Arbitrary Constants = Order of DE

Formation Strategy — Systematic Elimination of Constants

Practice Questions

Related Study Material

Practice Questions

Topic Information

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Track Your Learning

Study Tips

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Target Subjects

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Subject Mastery

Differential Equations — Order, Degree & Formation

1. Order and Degree of a Differential Equation

2. General Solution vs Particular Solution

3. Formation of a Differential Equation

Example 1 — One Constant: y=Ae2x

Example 2 — Two Constants: y=Asin⁡x+Bcos⁡x

Example 3 — Two Constants: y = Ax² + Bx

Example 4 (JEE Level) — Family of circles: x² + y² = r²

Example 5 (JEE Advanced Level) — y² = 4a(x − a)

4. Number of Arbitrary Constants = Order of DE

Formation Strategy — Systematic Elimination of Constants

Practice Questions

Related Study Material

Practice Questions

Topic Information

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Track Your Learning

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We Value Your Privacy

Example 1 — One Constant: $y = A e^{2 x}$

Example 2 — Two Constants: $y = A \sin x + B \cos x$