How much of JEE Main comes from Differential Equations?

JEEGenie estimates about 4% of JEE Main marks come from Differential Equations, based on recent papers. The share varies from year to year.

Which formulas matter most in Differential Equations?

The Differential Equations sheet on this page has 19 formula entries, starting with Variable Separable Method, Homogeneous Differential Equation, Linear Differential Equation (First Order), Bernoulli's Equation, Exact Differential Equation. Each one comes with when to use it and the trap to avoid.

What topics does Differential Equations cover for JEE?

Differential Equations covers Order and Degree, Variable Separable Equations, Linear Differential Equations, Homogeneous Equations, Formation of Differential Equations, Applications.

How should I revise Differential Equations before the exam?

Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Differential Equations questions on JEEGenie.

How many previous-year questions are there on Differential Equations?

JEEGenie has 350 previous-year questions on Differential Equations, and this page shows a few with worked explanations.

Is practice on Differential Equations free on JEEGenie?

Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.

Differential Equations for JEE Main & Advanced — formulas, what is asked, and traps

JEEGenie estimates about 4% of JEE Main marks come from Differential Equations. This page covers 19 formula entries, each with when to use it, what JEE asks and the traps. A common slip: Don't confuse order (highest derivative) with degree (power of highest derivative after removing radicals).

Key points to know

Differential Equation (DE). An equation involving derivatives of a dependent variable w.r.t. one or more independent variables.
Order and Degree of a DE. Order = highest derivative present. Degree = power of highest derivative after making DE polynomial in derivatives.
Solution of a DE. A function $y = f (x)$ that satisfies the DE identically. General solution has arbitrary constants; particular solution has specific values.
Initial Value Problem (IVP). A DE with conditions specifying the value of the dependent variable and/or its derivatives at a point.
Wronskian and Linear Independence. For two functions $y_{1}, y_{2}$ , Wronskian $W = y_{1} y_{2}^{'} - y_{2} y_{1}^{'}$ . If $W \neq 0$ , they are linearly independent.

Formula sheet

Name	Formula	When to use
Variable Separable Method	$\int f (x) d x = \int g (y) d y + C$	When DE can be written as $\frac{d y}{d x} = h (x) \cdot k (y)$ .
Homogeneous Differential Equation	$y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$	When DE is of the form $\frac{d y}{d x} = \frac{P (x, y)}{Q (x, y)}$ where P and Q are homogeneous of same degree.
Linear Differential Equation (First Order)	IF = $e^{\int P d x}$ , solution: $y \cdot I F = \int Q \cdot I F d x + C$	When DE is linear in y and its derivative, i.e., no $y^{2}$ or $y y^{'}$ terms.
Bernoulli's Equation	$v = y^{1 - n} \Rightarrow \frac{d v}{d x} + (1 - n) P v = (1 - n) Q$	When DE has a $y^{n}$ term on RHS, with n ≠ 0,1.
Exact Differential Equation	Condition: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ ; Solution: $\int M d x + \int (N - \frac{\partial}{\partial y} \int M d x) d y = C$	When DE is in differential form $M d x + N d y = 0$ and condition holds.
Integrating Factor for Non-Exact DE	IF = $e^{\int f (x) d x}$ or $e^{\int g (y) d y}$	When DE is not exact but one of the conditions for IF holds.
Orthogonal Trajectories	For family $F (x, y, c) = 0$ , find DE, then replace $y^{'}$ with $- 1 / y^{'}$ .	Finding curves perpendicular to a given family (e.g., circles, parabolas).
Solving DEs of Form $\frac{d^{2} y}{d x^{2}} = f (x)$	$y = \int (\int f (x) d x) d x + C_{1} x + C_{2}$	When the highest derivative is a function of x only.
Reducible to Variable Separable: $\frac{d y}{d x} = f (a x + b y + c)$	$v = a x + b y + c \Rightarrow \frac{d v}{d x} = a + b \frac{d y}{d x}$	When DE is of the form $\frac{d y}{d x} = f (a x + b y + c)$ .
General Solution of Linear DE with Constant Coefficients	Roots $m_{1}, m_{2}$ : real distinct $\Rightarrow y = C_{1} e^{m_{1} x} + C_{2} e^{m_{2} x}$ ; real equal $\Rightarrow y = (C_{1} + C_{2} x) e^{m_{1} x}$ ; complex $α \pm i β \Rightarrow y = e^{α x} (C_{1} \cos β x + C_{2} \sin β x)$	Solving homogeneous linear DEs with constant coefficients (order 2 or higher).
Particular Solution by Method of Undetermined Coefficients	If RHS = $e^{k x}$ , guess $A e^{k x}$ ; if $\sin k x$ or $\cos k x$ , guess $A \cos k x + B \sin k x$ ; if polynomial, guess polynomial of same degree.	When RHS is a combination of polynomials, exponentials, sines, cosines.
Particular Solution by Variation of Parameters	$y_{p} = - y_{1} \int \frac{y_{2} R}{W} d x + y_{2} \int \frac{y_{1} R}{W} d x$	When RHS is not of the form for undetermined coefficients, or for variable coefficient DEs.
Cauchy-Euler Equation	Roots $m_{1}, m_{2}$ : real distinct $\Rightarrow y = C_{1} x^{m_{1}} + C_{2} x^{m_{2}}$ ; real equal $\Rightarrow y = (C_{1} + C_{2} \ln x) x^{m_{1}}$ ; complex $α \pm i β \Rightarrow y = x^{α} (C_{1} \cos (β \ln x) + C_{2} \sin (β \ln x))$	When DE has terms like $x^{2} y^{''}$ , $x y^{'}$ , $y$ with constant coefficients.
Solving DEs Using Substitution $y = v x$ (Homogeneous) or $x = v y$	$y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$	When DE is homogeneous of degree 0.
Reduction of Order	$y_{2} = y_{1} \int \frac{e^{- \int P d x}}{y_{1}^{2}} d x$	When one solution is known for a second order linear DE.
Wronskian and Linear Independence	$W (y_{1}, y_{2}) = ∣ \begin{array}{cc} y_{1} & y_{2} \\ y_{1}^{'} & y_{2}^{'} \end{array} ∣$	Checking linear independence of solutions.
Solving DEs by Inspection (Special Forms)	$d (x y) = x d y + y d x$ , $d (\frac{y}{x}) = \frac{x d y - y d x}{x^{2}}$ , $d (\frac{x}{y}) = \frac{y d x - x d y}{y^{2}}$	When DE can be rearranged into exact differential forms.
Applications: Growth and Decay Problems	$N (t) = N_{0} e^{k t}$	Problems involving population growth, radioactive decay, Newton's law of cooling.
Applications: Newton's Law of Cooling	$T (t) = T_{s} + (T_{0} - T_{s}) e^{- k t}$	Problems where an object cools or heats to ambient temperature.

Open the Differential Equations formula page

What JEE asks

Differential Equation (DE). JEE often asks to find order and degree from a given DE, sometimes with radicals to test degree definition.
Order and Degree of a DE. JEE frequently tests degree after rationalization, e.g., ${(\frac{d^{2} y}{d x^{2}})}^{3} + \sqrt{y^{'}} = 0$ .
Solution of a DE. JEE may ask to verify if a given function is a solution, or find constants from conditions.
Variable Separable Method. JEE often gives DEs like $\frac{d y}{d x} = \frac{x}{y}$ or $\frac{d y}{d x} = e^{x + y}$ which are separable.
Homogeneous Differential Equation. JEE tests with functions like $\frac{d y}{d x} = \frac{x^{2} + y^{2}}{x y}$ or $\frac{d y}{d x} = \frac{y}{x} + \sin (\frac{y}{x})$ .
Linear Differential Equation (First Order). JEE often gives DEs like $\frac{d y}{d x} + y \tan x = \sin x$ or $x \frac{d y}{d x} + 2 y = x^{2}$ .
Bernoulli's Equation. JEE tests with $\frac{d y}{d x} + \frac{y}{x} = y^{2}$ or $\frac{d y}{d x} + y = x y^{3}$ .
Exact Differential Equation. JEE may ask to find integrating factor or solve exact DEs like $(x^{2} + y^{2}) d x + 2 x y d y = 0$ .
Integrating Factor for Non-Exact DE. JEE often gives DEs like $(x^{2} + y^{2} + x) d x + x y d y = 0$ requiring IF.
Orthogonal Trajectories. JEE may ask for orthogonal trajectories of $x^{2} + y^{2} = a^{2}$ or $y = c x^{2}$ .
Formation of DE by Eliminating Arbitrary Constants. JEE tests with families like $y = A e^{x} + B e^{- x}$ (2 constants, so 2nd order DE).
Initial Value Problem (IVP). JEE often gives IVPs like $\frac{d y}{d x} = 2 x, y (0) = 1$ to find particular solution.
Solving DEs of Form $\frac{d^{2} y}{d x^{2}} = f (x)$ . JEE may give $y^{''} = \sin x$ with $y (0) = 0, y^{'} (0) = 1$ .
Reducible to Variable Separable: $\frac{d y}{d x} = f (a x + b y + c)$ . JEE tests with $\frac{d y}{d x} = (x + y)^{2}$ or $\frac{d y}{d x} = \sin (x - y)$ .
General Solution of Linear DE with Constant Coefficients. JEE often gives $y^{''} - 3 y^{'} + 2 y = 0$ or $y^{''} + 4 y = 0$ .
Particular Solution by Method of Undetermined Coefficients. JEE tests with $y^{''} + y = \sin x$ (guess $x (A \cos x + B \sin x)$ ) or $y^{''} - y = e^{x}$ .
Particular Solution by Variation of Parameters. JEE may ask for $y^{''} + y = \sec x$ or $y^{''} - 2 y^{'} + y = e^{x} \ln x$ .
Cauchy-Euler Equation. JEE tests with $x^{2} y^{''} - 2 x y^{'} + 2 y = 0$ or $x^{2} y^{''} + x y^{'} + y = 0$ .
Solving DEs Using Substitution $y = v x$ (Homogeneous) or $x = v y$ . JEE may give $\frac{d y}{d x} = \frac{x^{2} + y^{2}}{2 x y}$ or $\frac{d y}{d x} = \frac{y}{x} + \tan (\frac{y}{x})$ .
Reduction of Order. JEE may ask to find second solution of $x^{2} y^{''} - 2 y = 0$ given $y_{1} = x^{2}$ .
Wronskian and Linear Independence. JEE may ask to compute Wronskian of $e^{x}$ and $e^{- x}$ .
Solving DEs by Inspection (Special Forms). JEE tests with $x d y + y d x = 0$ (gives $x y = C$ ) or $\frac{x d y - y d x}{x^{2}} = 0$ .
Applications: Growth and Decay Problems. JEE gives problems like 'A population doubles in 10 years, find when it triples'.
Applications: Newton's Law of Cooling. JEE tests with 'A body cools from 60°C to 50°C in 10 min, find time to reach 40°C'.

Common traps

Don't confuse order (highest derivative) with degree (power of highest derivative after removing radicals). Differential Equation (DE)
If DE involves radicals like $\sqrt{y^{'}}$ , first square to remove radical before finding degree. Order and Degree of a DE
General solution of an nth order DE has n independent arbitrary constants. Solution of a DE
Don't forget the constant of integration; it's crucial for general solution. Variable Separable Method
First check homogeneity: each term in numerator and denominator of $f$ must have same degree. Homogeneous Differential Equation
Ensure DE is in standard form; coefficient of $\frac{d y}{d x}$ must be 1. Linear Differential Equation (First Order)
For n=0 or n=1, it's already linear or separable; don't use Bernoulli substitution. Bernoulli's Equation
Check exactness condition before solving; if not exact, look for integrating factor. Exact Differential Equation
Compute partial derivatives correctly; common sign errors. Integrating Factor for Non-Exact DE
First find DE of given family (eliminate constant), then replace slope. Orthogonal Trajectories
Number of differentiations = number of arbitrary constants to eliminate. Formation of DE by Eliminating Arbitrary Constants
Use conditions to find particular constants; don't leave general solution. Initial Value Problem (IVP)
Don't forget the two constants of integration. Solving DEs of Form $\frac{d^{2} y}{d x^{2}} = f (x)$
After substitution, solve for $\frac{d y}{d x}$ in terms of $v$ correctly. Reducible to Variable Separable: $\frac{d y}{d x} = f (a x + b y + c)$

Topics in Differential Equations

Order and Degree
Variable Separable Equations
Linear Differential Equations
Homogeneous Equations
Formation of Differential Equations
Applications

Previous-year questions on Differential Equations

JEEGenie has 350 previous-year questions on Differential Equations. A few worked examples follow.

Year	Questions
2026	21
2025	23
2024	40
2023	25
2022	41
2021	44
2020	17
2019	39
2018	9
2017	6
2016	9
2015	7
2014	7
2013	11
2012	6
2011	4
2010	1
2009	3
2008	2
2007	4
2006	2
2005	4
2004	4
2003	6
2002	2
2001	1
2000	1
1999	4
1998	1
1997	4
1996	1
1995	1

Question 1 medium

If the curve

y = f (x)

passes through the point

(1, e)

and satisfies the differential equation

d y = y (2 + l o g_{e} x) d x

x > 0

, then

f (e)

is equal to :

$e^{e}$
$e^{e^{2}}$
$e^{2 e}$
$e^{2^{e}}$

Answer: C

Explanation.

\frac{d y}{y} = (2 + \log_{e} x) d x

Integrate both sides:

\ln y = 2 x + \int \log_{e} x d x

\int \log_{e} x d x = x \log_{e} x - x

So,

\ln y = 2 x + x \log_{e} x - x + C = x + x \log_{e} x + C

Given

y (1) = e

\ln e = 1 + 1 \cdot 0 + C \Rightarrow 1 = 1 + C \Rightarrow C = 0

Thus,

\ln y = x + x \log_{e} x = x (1 + \log_{e} x)

y = e^{x (1 + \log_{e} x)}

Now find

f (e)

f (e) = e^{e (1 + \log_{e} e)} = e^{e (1 + 1)} = e^{2 e}

C

From NTA/IIT papers: JEE Main 2026 · 2 Apr Shift 1

Question 2 hard

Let

g

be a differentiable function such that

\int_{0}^{x} g (t) d t = x - \int_{0}^{x} tg (t) d t, x \geq 0

and let

y = y (x)

satisfy the differential equation

\frac{d y}{d x} - y \tan x = 2 (x + 1) \sec x g (x), x \in [0, \frac{π}{2})

. If

y (0) = 0

, then

y (\frac{π}{3})

is equal to

$\frac{4 π}{3}$
$\frac{2 π}{3}$
$\frac{2 π}{3 \sqrt{3}}$
$\frac{4 π}{3 \sqrt{3}}$

Answer: A

Explanation. Differentiate the given integral equation:

g (x) = 1 - x g (x) \Rightarrow g (x) (1 + x) = 1 \Rightarrow g (x) = \frac{1}{1 + x}

. The differential equation is

\frac{d y}{d x} - y \tan x = 2 (x + 1) \sec x \cdot \frac{1}{1 + x} = 2 \sec x

. Integrating factor:

μ = e^{- \int \tan x d x} = e^{\ln (\cos x)} = \cos x

. Multiply:

\frac{d}{d x} (y \cos x) = 2 \sec x \cdot \cos x = 2

. Integrate:

y \cos x = 2 x + C

y (0) = 0 \Rightarrow 0 = 0 + C \Rightarrow C = 0

. Thus

y = \frac{2 x}{\cos x}

. At

x = \frac{π}{3}

y (\frac{π}{3}) = \frac{2 \cdot \frac{π}{3}}{\frac{1}{2}} = \frac{4 π}{3}

. The answer is option A.

From NTA/IIT papers: JEE Main 2025 · 3 Apr Shift 1

Question 3 easy

The temperature

T (t)

of a body at time

t = 0

160^{\circ} F

and it decreases continuously as per the differential equation

\frac{d T}{d t} = - K (T - 80)

, where

K

is a positive constant. If

T (15) = 120^{\circ} F

, then

T (45)

is equal to

90 $^{\circ}$ F
85 $^{\circ}$ F
80 $^{\circ}$ F
95 $^{\circ}$ F

Answer: A

Explanation. Newton's law of cooling gives the differential equation

\frac{d T}{d t} = - K (T - 80)

Separate variables:

\int \frac{d T}{T - 80} = - K \int d t

\ln ∣ T - 80 ∣ = - K t + C

T - 80 = A e^{- K t}

t = 0

T = 160

160 - 80 = A \Rightarrow A = 80

T (t) = 80 + 80 e^{- K t}

t = 15

T = 120

120 = 80 + 80 e^{- 15 K}

40 = 80 e^{- 15 K}

e^{- 15 K} = \frac{1}{2}

Now at

t = 45

T (45) = 80 + 80 e^{- 45 K}

Since

e^{- 45 K} = (e^{- 15 K})^{3} = {(\frac{1}{2})}^{3} = \frac{1}{8}

T (45) = 80 + 80 \cdot \frac{1}{8} = 80 + 10 = 90^{\circ} F

The other options fail because they do not satisfy the exponential decay relation

T - 80 \propto e^{- K t}

. Answer: (A)

From NTA/IIT papers: JEE Main 2024 · 31 Jan Shift 2

Question 4 medium

Let

y = y (x)

be a solution of the differential equation

(x \cos x) d y + (x y \sin x + y \cos x - 1) d x = 0, 0 < x < \frac{π}{2}

. If

\frac{π}{3} y (\frac{π}{3}) = \sqrt{3}

, then

∣ \frac{π}{6} y^{''} (\frac{π}{6}) + 2 y^{'} (\frac{π}{6}) ∣

is equal to ____________.

Answer: 2

Explanation. Rewrite the equation as

x \cos x \frac{d y}{d x} + y (x \sin x + \cos x) = 1 \Rightarrow \frac{d y}{d x} + y (\tan x + \frac{1}{x}) = \frac{1}{x \cos x} .

**Integrating factor:**

e^{\int (\tan x + 1 / x) d x} = x \sec x

. Then

\frac{d}{d x} (x y \sec x) = x \sec x \cdot \frac{1}{x \cos x} = \sec^{2} x \Rightarrow x y \sec x = \tan x + C .

x = \frac{π}{3}

\frac{π}{3} y (\frac{π}{3}) = \sqrt{3}

and

\sec \frac{π}{3} = 2

, so

2 \sqrt{3} = \sqrt{3} + C

and

C = \sqrt{3}

. Hence

x y = \sin x + \sqrt{3} \cos x .

**Differentiate twice.** Let

u = x y

u^{'} = y + x y^{'} = \cos x - \sqrt{3} \sin x,

u^{''} = 2 y^{'} + x y^{''} = - \sin x - \sqrt{3} \cos x .

The required expression is

x y^{''} + 2 y^{'} = u^{''}

. At

x = \frac{π}{6}

\frac{π}{6} y^{''} ⁣ (\frac{π}{6}) + 2 y^{'} ⁣ (\frac{π}{6}) = - \frac{1}{2} - \sqrt{3} \cdot \frac{\sqrt{3}}{2} = - \frac{1}{2} - \frac{3}{2} = - 2.

Its absolute value is

2

. Answer: 2

From NTA/IIT papers: JEE Main 2023 · 6 Apr Shift 1

Question 5 hard

y (x)

is the solution of the differential equation

x d y - (y^{2} - 4 y) d x = 0

for

x > 0

y (1) = 2

, and the slope of the curve

y = y (x)

is never zero, then the value of

10 y (\sqrt{2})

is _____.

Answer: 8

Explanation. We have

x d y - (y^{2} - 4 y) d x = 0 ⟹ \frac{d y}{d x} = \frac{y^{2} - 4 y}{x} .

Separate variables:

\frac{d y}{y (y - 4)} = \frac{d x}{x} .

Partial fractions:

\frac{1}{y (y - 4)} = \frac{1}{4} (\frac{1}{y - 4} - \frac{1}{y}) .

\frac{1}{4} \ln ∣ \frac{y - 4}{y} ∣ = \ln x + C .

Given

y (1) = 2

\frac{1}{4} \ln ∣ \frac{2 - 4}{2} ∣ = \frac{1}{4} \ln 1 = 0 ⟹ C = 0.

Thus

\ln ∣ \frac{y - 4}{y} ∣ = 4 \ln x = \ln x^{4},

∣ \frac{y - 4}{y} ∣ = x^{4} .

Since slope is never zero,

y \neq 0, 4

. At

x = 1

y = 2

, so

\frac{y - 4}{y} = - 1

, hence the sign is negative:

\frac{y - 4}{y} = - x^{4} .

Solve:

y - 4 = - x^{4} y ⟹ y (1 + x^{4}) = 4 ⟹ y = \frac{4}{1 + x^{4}} .

Now

y (\sqrt{2}) = \frac{4}{1 + (\sqrt{2})^{4}} = \frac{4}{1 + 4} = \frac{4}{5} .

Therefore

10 y (\sqrt{2}) = 10 \cdot \frac{4}{5} = 8.

8

From NTA/IIT papers: JEE Advanced 2022

Frequently asked questions

How much of JEE Main comes from Differential Equations?: JEEGenie estimates about 4% of JEE Main marks come from Differential Equations, based on recent papers. The share varies from year to year.
Which formulas matter most in Differential Equations?: The Differential Equations sheet on this page has 19 formula entries, starting with Variable Separable Method, Homogeneous Differential Equation, Linear Differential Equation (First Order), Bernoulli's Equation, Exact Differential Equation. Each one comes with when to use it and the trap to avoid.
What topics does Differential Equations cover for JEE?: Differential Equations covers Order and Degree, Variable Separable Equations, Linear Differential Equations, Homogeneous Equations, Formation of Differential Equations, Applications.
How should I revise Differential Equations before the exam?: Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Differential Equations questions on JEEGenie.
How many previous-year questions are there on Differential Equations?: JEEGenie has 350 previous-year questions on Differential Equations, and this page shows a few with worked explanations.
Is practice on Differential Equations free on JEEGenie?: Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.