JEE (Main) · Mathematics, Class 12
Differential Equations: JEE Main PYQ, 2020–2025
Differential Equations (Mathematics, Class 12) has been asked 174 times in the 111 JEE (Main) shift papers from 2020–2025: about 1.6 questions a paper, 6% of all Mathematics questions. It is the 7th most-asked of 19 Mathematics chapters. The 2025 shifts had 21 from it in all. Last checked 28 October 2026.
Questions per year, all shifts
Year by year.
| Year | Questions |
|---|---|
| 2020 | 16in 14 papers |
| 2021 | 43in 26 papers |
| 2022 | 44in 24 papers |
| 2023 | 14in 12 papers |
| 2024 | 36in 18 papers |
| 2025 | 21in 17 papers |
JEE (Main) is sat in many shifts a year, each with its own paper; the counts add up every shift. 2021–2024 papers had 30 questions a subject, the others 25.
Inside the chapter
Topics asked most.
- Linear differential equations87
- Separation of variables61
- Homogeneous equations19
- Order and degree7
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10 Differential Equations questions to practise.
Our own questions, written in the JEE Main pattern and checked, each with an explanation. Pick an option to see the answer.
Q1 · Mathematics · Differential Equations
For x > 0, what is the general solution of dy/dx − y/x = x?
Answer: (A)
P = −1/x, so the integrating factor is e^{−ln x} = 1/x. Multiplying gives (1/x) dy/dx − y/x² = 1, which is d(y/x)/dx = 1. Integrating gives y/x = x + C, so y = x² + Cx.
Q2 · Mathematics · Differential Equations
The solution of dy/dx = e^(x+y) with y(0) = 0 is defined on an interval of the form (−∞, x₀). What is x₀?
Answer: (A)
Write dy/dx = e^x · e^y and separate: e^(−y) dy = e^x dx. Integrating gives −e^(−y) = e^x + C. Using y(0) = 0 gives −1 = 1 + C, so C = −2. Hence e^(−y) = 2 − e^x, and y = −ln(2 − e^x). This is defined only when 2 − e^x > 0, i.e. x < ln 2. As x → ln 2 from the left, y → +∞, so the solution blows up there.
Q3 · Mathematics · Differential Equations
Let y(x) be the solution of (1 + x^2) dy/dx = 1 + y^2 with y(0) = 1. What is the limit of y(x) as x tends to -infinity?
Answer: (A)
Separate the variables: dy/(1 + y^2) = dx/(1 + x^2). Integrating gives arctan y = arctan x + c. At (0, 1), arctan 1 = pi/4 = c. So arctan y = arctan x + pi/4, which gives y = tan(arctan x + pi/4) = (1 + x)/(1 - x) for x < 1. As x tends to -infinity, arctan x tends to -pi/2, so arctan y tends to -pi/4 and y tends to tan(-pi/4) = -1. The rational form agrees: (1 + x)/(1 - x) tends to -1.
Q4 · Mathematics · Differential Equations
For −π/2 < x < π/2, what is the general solution of dy/dx + y tan x = sec x?
Answer: (A)
P = tan x, so ∫P dx = ln|sec x| and the integrating factor is sec x. Multiplying gives d(y sec x)/dx = sec² x, so y sec x = tan x + C. Multiply by cos x: y = sin x + C cos x.
Q5 · Mathematics · Differential Equations
y is the solution of dy/dx + y = 1 with y(0) = 0. What is y(ln 2)?
Answer: (A)
With integrating factor eˣ: d(y eˣ)/dx = eˣ, so y eˣ = eˣ + C. y(0) = 0 gives C = −1, so y = 1 − e^(−x). Then y(ln 2) = 1 − 1/2 = 1/2.
Q6 · Mathematics · Differential Equations
The family of all parabolas whose axis is parallel to the x-axis is given by (y − k)² = 4a(x − h), where a, h and k are arbitrary constants. The differential equation of this family, free of a, h and k, has order m and degree n. What is (m, n)?
Answer: (D)
Three arbitrary constants mean the order is 3. Differentiating (y − k)² = 4a(x − h) gives (y − k)y' = 2a. Differentiating again gives y'² + (y − k)y'' = 0. Differentiating a third time gives 3y'y'' + (y − k)y''' = 0. From the second relation y − k = −y'²/y''. Substituting: 3y'y'' − y'²y'''/y'' = 0, so 3(y'')² = y'y''' after dividing by y'. The highest-order derivative y''' appears to the first power, so the degree is 1 even though y'' is squared. The pair is (3, 1).
Q7 · Mathematics · Differential Equations
Let y(x) satisfy dy/dx = -y² with y(0) = 1. What is the value of y(3)?
Answer: (A)
Separate the variables as dy/y² = -dx. Integrating gives -1/y = -x + C, so 1/y = x + C'. Using y(0) = 1 gives C' = 1, so y = 1/(x + 1). Then y(3) = 1/4.
Q8 · Mathematics · Differential Equations
A curve passes through (1, 3). At every point P(x, y) on it with x ≠ 0, the tangent at P cuts the x-axis at a point whose abscissa is half the abscissa of P. What is the ordinate of the point on the curve with abscissa 2?
Answer: (A)
The tangent at (x, y) is Y − y = (dy/dx)(X − x). Putting Y = 0 gives the x-intercept X = x − y/(dy/dx). The condition says X = x/2, so y/(dy/dx) = x/2 and dy/dx = 2y/x. Separate: dy/y = 2 dx/x, so ln y = 2 ln x + c and y = Cx². Since y(1) = 3, C = 3 and y = 3x². Therefore y(2) = 12.
Q9 · Mathematics · Differential Equations
The curve y = y(x), x > 0, satisfies dy/dx = (x + y)/(x − y) and passes through (1, 0). At what x-coordinate does this curve meet the line y = x?
Answer: (A)
The equation is homogeneous of degree 0, so put y = vx. Then v + x dv/dx = (1 + v)/(1 − v), which gives x dv/dx = (1 + v²)/(1 − v). Separating variables, ∫(1 − v)/(1 + v²) dv = ∫dx/x, so tan⁻¹v − ½ ln(1 + v²) = ln x + C. Since −½ ln(1 + v²) = −½ ln(x² + y²) + ln x, the ln x terms cancel and tan⁻¹(y/x) = ½ ln(x² + y²) + C. The point (1, 0) gives C = 0. On y = x we have tan⁻¹1 = π/4, so π/4 = ½ ln(2x²). Hence 2x² = e^(π/2) and, for x > 0, x = e^(π/4)/√2.
Q10 · Mathematics · Differential Equations
A population P(t) grows at a rate proportional to its current size, so dP/dt = kP with k > 0. If the population doubles in 5 years, by what factor has it increased after 15 years?
Answer: (D)
Separating dP/P = k dt gives P = P₀e^(kt). Doubling in 5 years means e^(5k) = 2. At t = 15, P/P₀ = e^(15k) = (e^(5k))³ = 2³ = 8.
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All 174 real JEE Main questions from Differential Equations.
The questions from the actual papers, with the official answer and an explanation for each — chapter by chapter, or as full timed papers.
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About Differential Equations in JEE Main.
How many questions come from Differential Equations in JEE Main?
Differential Equations (Mathematics, Class 12) has been asked 174 times in the 111 JEE (Main) shift papers from 2020–2025: about 1.6 questions a paper, 6% of all Mathematics questions. It is the 7th most-asked of 19 Mathematics chapters. The 2025 shifts had 21 from it in all.
Which topics of Differential Equations are asked most in JEE Main?
Linear differential equations (87), Separation of variables (61), Homogeneous equations (19), Order and degree (7), counted over the papers from 2020–2025.
Is Differential Equations important for JEE Main 2027?
It is in the syllabus in force, so it can be asked. By past papers it ranks 7 of 19 in Mathematics; use that to decide how much time it gets, not whether to study it.
Where can I solve Differential Equations JEE Main previous year questions?
On Nikaalle, free: sign up and practise every past JEE Main question from Differential Equations with an explanation for each, or sit the full papers timed. The ten questions on this page are our own, written in the same pattern.
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