If you are wondering how to study Differential Equations for JEE, start with one number. JEE Main asked 79 questions from this chapter across 2024, 2025 and 2026, and almost half of them, 39, were first-order linear equations solved with an integrating factor. JEE Advanced asked 15 questions from the chapter between 2015 and 2026. This guide breaks all of them down by method, based on the papers conducted by the National Testing Agency (jeemain.nta.ac.in) for JEE Main and the JEE Advanced papers from 2015 to 2026.
We went through every one of those questions and tagged it by the method that solves it: linear equations, variables separable, homogeneous equations, exact or total-differential forms, functional and integral equations that turn into a differential equation, and forming equations or applications. The tables below show exactly where the marks come from, and the rest of the guide shows how to win them.
This chapter is unusual because it is method-driven, not concept-driven. Almost every question gives you an equation and asks for a value of the solution. Recognise the type in the first ten seconds and the rest is integration. Misread the type and no amount of algebra will save the question.
It gives you one or two questions in nearly every JEE Main shift. JEE Main had 37 questions from this chapter in 2024, 21 in 2025 and 21 in 2026. That is 6.17%, 4.42% and 4.42% of all Maths questions in those years, or roughly 1.1 to 1.9 questions per paper.
One method carries half the chapter. The integrating-factor method for dy/dx + P(x)y = Q(x) solved 39 of the 79 Main questions. In 2025 it solved 15 of the 21. Very few chapters let a single technique deliver that many marks.
The syllabus inside the chapter has narrowed. Forming a differential equation from a family of curves, and word problems such as Newton's law of cooling, gave 4 questions in 2024 and none in 2025 or 2026. The marks have moved to solving equations, often with a substitution first. Your preparation should move with them.
It is the last step of Calculus. Every question needs clean integration, and many end with a definite integral, an area or a maximum value. A strong Differential Equations chapter is also a revision of Indefinite and Definite Integration.
"In this chapter, the first ten seconds decide the question. Before you touch the equation, say out loud what type it is: linear, separable, homogeneous or exact. Students who start manipulating before naming the type go in circles." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
Real result · JEE 2026
Half of this chapter's JEE Main questions are one method done cleanly. Learn it 1-on-1 with an IITian Maths HOD and walk into the exam knowing the type of every equation in ten seconds.
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Any plan for how to study Differential Equations for JEE should start from the papers. Here is how many questions JEE Main asked from this chapter in each of the last three years.
| Year | Questions | MCQ + Numerical | Share of Maths Qs |
|---|---|---|---|
| 2024 | 37 | 23 + 14 | 6.17% |
| 2025 | 21 | 16 + 5 | 4.42% |
| 2026 | 21 | 15 + 6 | 4.42% |
| Total | 79 | 54 + 25 | — |
And here are the same 79 questions sorted by the method that solves them.
| Method | 2024 | 2025 | 2026 | Total |
|---|---|---|---|---|
| Linear first-order (integrating factor; includes equations linear in x and Bernoulli-type equations that become linear) | 16 | 15 | 8 | 39 |
| Variables separable (includes substitutions like v = x + y or v = 2x + 3y) | 8 | 1 | 4 | 13 |
| Functional and integral equations that reduce to a differential equation | 1 | 4 | 4 | 9 |
| Homogeneous equations (y = vx) | 4 | 1 | 3 | 8 |
| Exact and total-differential forms | 2 | 0 | 1 | 3 |
| Forming equations and applications (family of curves, cooling, curve geometry) | 4 | 0 | 0 | 4 |
| Not classified* | 2 | 0 | 1 | 3 |
| Total | 37 | 21 | 21 | 79 |
*The archived text of three questions did not render fully, so we could not tell which method they need. They are counted in the yearly totals only. Questions that need two methods (for example, a substitution followed by an integrating factor) are counted under the method that finishes them.
The message of this table is simple. Linear equations gave 39 of 79 questions. Add variables separable and homogeneous equations and you have 60 of 79, about three in every four. Functional and integral equations are the one group that is growing: 1 in 2024, then 4 in each of 2025 and 2026.
Pattern 1: Linear equations with an awkward P(x). JEE Main rarely gives a clean dy/dx + y = x. It gives (x2 + 4)2 dy + (2x3y + 8xy − 2) dx = 0 (2024), (x2 + 1)y′ − 2xy = (x4 + 2x2 + 1) cos x (2025), or a 2026 equation where P(x) was a long fraction in x3 and e−2x. In each case, dividing to get the standard form shows that ∫P dx turns out to be a logarithm, so the integrating factor is a polynomial or a simple product.
Pattern 2: Equations that are linear in x, not y. y2 dx + (x − 1/y) dy = 0 (2025), (1 + y2) + (x − 2etan−1y) dy/dx = 0 (2025) and dx/dy = (1 + x − y2)/y (2024) all become easy once you write dx/dy + P(y)x = Q(y). If an equation looks impossible in dy/dx, flip it.
Pattern 3: A substitution first, then a linear equation. 2026 asked xy′ − sin 2y = x3(2 − x3) cos2y, which becomes linear after dividing by cos2y and putting t = tan y. 2024 asked sec y · y′ + 2x sin y = x3 cos y (divide by cos y and put t = tan y), and dy/dx = 2x(x + y)3 − x(x + y) − 1 (put v = x + y). Spotting the substitution is the whole question.
Pattern 4: Integral and functional equations. 2025 asked ∫0x t f(t) dt = x2 f(x) with f(2) = 3, and a functional equation f(x + y) = f(x)f(y) combined with a second-order condition. 2026 asked f2(x) = 25 + ∫0x (f2 + f′2) dt. Differentiate once, and each becomes a first-order equation you already know how to solve.
Pattern 5: The question does not stop at y(x). Many questions solve the equation and then ask for something else: the local maximum of the solution (2026, an answer of 16), an area between the solution curve and a line (2024), a definite integral of the solution (2025) or the maximum and minimum of y on an interval (2024). Leave time for that last step.
This is the 12-year view that most guides skip. JEE Advanced asked this chapter in 10 of the 12 years from 2015 to 2026, missing only 2020 and 2021.
| Year | Qs | What was asked |
|---|---|---|
| 2026 | 2 | A linear equation whose left side is the derivative of a quotient, then local maximum, local minimum and monotonicity of the solution (multi-correct) · a separable equation for a positive solution y, ending in a quadratic (single correct) |
| 2025 | 2 | x2y′ + xy = x2 + y2 with y(1) = 0, a homogeneous equation (numerical) · three functions y1, y2, y3 defined by linear equations, combined in a limit (numerical) |
| 2024 | 1 | A limit condition limt→x (t10f(x) − x10f(t))/(t − x) = 0 that becomes a separable equation, f(1) = 2 (single correct) |
| 2023 | 2 | An integral equation in f that becomes a linear equation; find f(e) (single correct) · a linear equation with a given point; maximum value of the solution (numerical) |
| 2022 | 2 | dy/dx + 12y = cos(πx/12), y(0) = 0: statements about the solution (multi-correct) · a separable equation whose solution's slope is never zero (numerical) |
| 2019 | 1 | A curve through (1, 0) whose tangent at P meets the y-axis at YP with PYP = 1 (multi-correct) |
| 2018 | 1 | f(0) = 0 and f′ = (2 + 5f)(5f − 2): a separable equation, then a limit (numerical) |
| 2017 | 1 | A separable equation with nested square roots in x; find y(256) (single correct) |
| 2016 | 2 | A solution curve through (1, 3) of an equation that becomes homogeneous after a shift (multi-correct) · f′(x) = 2 − f(x)/x with f(1) ≠ 1 (multi-correct) |
| 2015 | 1 | (1 + ex)y′ + yex = 1 with y(0) = 2 (multi-correct) |
| Total | 15 | 6 multi-correct · 5 numerical · 4 single correct |
One 2017 question about g(x) = ∫ sin−1t dt between sin x and sin 2x, filed under this chapter in the archive, belongs to Definite Integration and is not counted.
Advanced follows the same method mix as Main: linear equations lead (6 of 15), then variables separable (4), homogeneous equations (2), functional or limit conditions (2) and one geometric application. The difference is in what comes after the solution. Advanced asks multi-correct statements about the solution's maximum, monotonicity, periodicity or limits, so solving the equation is only half the work.
The smartest way to decide how to study Differential Equations for JEE is to match your time to each method. Use the time column in your mock tests: if you are consistently slower than the target, that method needs more practice, not more reading.
| Method | Main Qs (2024–26) | Advanced Qs (2015–26) | Difficulty | Target time |
|---|---|---|---|---|
| Linear first-order | 39 | 6 | Medium | 2–3 min |
| Variables separable | 13 | 4 | Easy–Medium | 90 s–2 min |
| Functional and integral equations | 9 | 2 | Medium–Hard | 2–4 min |
| Homogeneous equations | 8 | 2 | Medium | 2–3 min |
| Exact and total-differential forms | 3 | 0 | Medium | 90 s–2 min |
| Forming equations and applications | 4 | 1 | Easy–Medium | 2 min |
Difficulty and target time are our faculty's guidance for exam practice, not exam data.
Linear first-order equations are dy/dx + P(x)y = Q(x), solved with the integrating factor e∫P dx. This group also includes equations linear in x and Bernoulli-type equations that become linear after a substitution.
Variables separable covers equations you can split into f(y) dy = g(x) dx, directly or after a substitution like v = x + y or v = ax + by + c.
Functional and integral equations give f through an integral or a relation like f(x + y) = f(x)f(y). Differentiating turns them into one of the other types.
Homogeneous equations are those where dy/dx is a function of y/x. Put y = vx and the equation separates. Some only become homogeneous after shifting the origin.
Exact and total-differential forms use known differentials like d(xy) = x dy + y dx and d(y/x) = (x dy − y dx)/x2.
Forming equations and applications means eliminating constants from a family of curves, or setting up an equation from a word problem or a tangent condition. It gave 4 Main questions in 2024 and none since.
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Linear, separable and homogeneous equations gave three of every four JEE Main questions from 2024 to 2026. Work through them with your own faculty, at your pace, with every doubt answered the same day.
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Knowing how to study Differential Equations for JEE also means knowing what not to do. These are the mistakes we see most often when students solve this chapter's PYQs.
Keep this table open while you practise. Most of the chapter runs on these results, which is why learning how to study Differential Equations for JEE is mostly about recognising the type and applying the right one quickly.
| Method | Formula / result |
|---|---|
| Linear in y | dy/dx + P(x)y = Q(x) · IF = e∫P dx · y · IF = ∫Q · IF dx + C |
| Linear in x | dx/dy + P(y)x = Q(y) · IF = e∫P dy · x · IF = ∫Q · IF dy + C |
| Bernoulli | dy/dx + Py = Qyn · divide by yn, put t = y1−n · then dt/dx + (1 − n)Pt = (1 − n)Q |
| Useful IFs | e∫(1/x)dx = x · e∫(−1/x)dx = 1/x · e∫tan x dx = sec x · e∫2x/(1+x2)dx = 1 + x2 · e∫dx/(1+x2) = etan−1x |
| Variables separable | f(y) dy = g(x) dx ⇒ ∫f(y) dy = ∫g(x) dx + C |
| Reducible to separable | dy/dx = f(ax + by + c): put v = ax + by + c, so dv/dx = a + b f(v) |
| Homogeneous | dy/dx = F(y/x): put y = vx, dy/dx = v + x dv/dx ⇒ dv/(F(v) − v) = dx/x |
| Shifted homogeneous | dy/dx = (a1x + b1y + c1)/(a2x + b2y + c2): put x = X + h, y = Y + k with (h, k) the intersection of the two lines |
| Exact differentials | x dy + y dx = d(xy) · (x dy − y dx)/x2 = d(y/x) · (x dy − y dx)/y2 = −d(x/y) · (x dy − y dx)/(x2 + y2) = d(tan−1(y/x)) |
| Exact differentials | (x dx + y dy)/(x2 + y2) = d(½ ln(x2 + y2)) · (x dy + y dx)/(xy) = d(ln xy) · xndy + nxn−1y dx = d(xny) |
| Integral equations | d/dx ∫ax f(t) dt = f(x) · put x = a in the original to get f(a) or another condition |
| Functional equations | f(x + y) = f(x)f(y), f differentiable ⇒ f′(x) = f′(0)f(x) ⇒ f(x) = ef′(0)x |
| Order and degree | Order = highest derivative · degree = its power after the equation is made polynomial in derivatives · n arbitrary constants ⇒ order n |
| Growth and cooling | dN/dt = kN ⇒ N = N0ekt · Newton's cooling: dT/dt = −k(T − Ts) ⇒ T − Ts = (T0 − Ts)e−kt |
| Curve geometry | Tangent at (x, y) meets the y-axis at (0, y − xy′) and the x-axis at (x − y/y′, 0) · length of tangent = |y|√(1 + y′2)/|y′| |
This is the core of how to study Differential Equations for JEE: one method at a time, starting with the ones that have given the most marks. Do not move on until you can solve the PYQ type listed under each heading without notes.
This is half the chapter, so it deserves half your practice time. The routine never changes:
Worked example from JEE Main 2024: (1 + x2)y′ + y = etan−1x with y(1) = 0. Divide by 1 + x2: P = 1/(1 + x2), so IF = etan−1x. Then y etan−1x = ∫e2tan−1x/(1 + x2) dx = ½e2tan−1x + C. From y(1) = 0, C = −½eπ/2, so y(0) = ½(1 − eπ/2).
Two extensions give many of the harder questions. First, equations linear in x: if the coefficient of dx contains y2 or etan−1y and x appears only to the first power, switch to dx/dy. Second, substitutions that create a linear equation: t = tan y, t = sin y, t = 1/y or v = x + y. The 2026 question xy′ − sin 2y = x3(2 − x3) cos2y is linear in t = tan y once you divide by cos2y.
Practice target: 30 linear-equation PYQs, including at least five linear in x and five that need a substitution first. Time each one; aim for under three minutes.
"The integrating factor is designed by the paper-setter. If your IF comes out ugly, you have made a mistake in P(x). In the actual exam the IF is almost always something simple: x, 1 + x2, sec x or etan−1x." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
Check for separation first, because it is the fastest method when it works.
Practice target: 15 PYQs from this group. For each, write the substitution (if any) in the margin before you begin.
This is the fastest-growing group. The method has three steps:
Worked example from JEE Main 2025: ∫0x t f(t) dt = x2f(x) with f(2) = 3. Differentiating, x f(x) = 2x f(x) + x2 f′(x), so x f′(x) = −f(x), which separates to f(x) = c/x. From f(2) = 3, c = 6, so f(6) = 1. JEE Advanced 2023 used the same idea with an integral from 1 to x, and JEE Advanced 2024 used a limit condition that becomes f′/f = 10/x.
For functional equations like f(x + y) = f(x)f(y), differentiate with respect to y and put y = 0. This gives f′(x) = f′(0)f(x), a separable equation with an exponential solution.
Practice target: 10 integral and functional equation PYQs. Always write the initial condition before you start solving.
"An integral equation is a differential equation in disguise. Differentiate it, and do not forget the free initial condition you get by putting x equal to the lower limit. That condition is the one most students miss." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
If every term in the numerator and denominator has the same degree, or dy/dx is a function of y/x only, put y = vx:
For equations like dy/dx = (x + y − 2)/(x − y) (2024), first shift the origin to the point where x + y − 2 = 0 and x − y = 0 meet, here (1, 1). In the new coordinates the equation is homogeneous.
Practice target: 10 homogeneous PYQs, including two with a shifted origin.
These are small groups, but each has a fixed trick:
Practice target: 8 PYQs across these three groups. Learn the exact-differential list by heart; it is the fastest way to finish these.
That is how to study Differential Equations for JEE method by method. Once you have finished all five groups, test yourself under exam conditions with our JEE test series, and bring any question that still does not make sense to the JEE Doubt Lab for a 1-on-1 explanation.
Ready for exam-level practice?
Integral equations and multi-correct statements about solutions are where most students lose marks here. Practise them 1-on-1 with an IITian Maths HOD who has taught them for years.
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If you are deciding how to study Differential Equations for JEE Advanced as well as Main, this section is for you. Advanced uses the same methods but goes further in four places.
6 of the 15 Advanced questions were multi-correct. Advanced 2022 asked whether the solution of dy/dx + 12y = cos(πx/12) is increasing, decreasing or periodic, and whether some horizontal line meets it at infinitely many points. Advanced 2026 asked about the local maximum, local minimum and monotonicity of a solution. You need Application of Derivatives ready the moment the equation is solved.
Advanced 2018 asked for lim f(x) as x → −∞ for the solution of f′ = (2 + 5f)(5f − 2) with f(0) = 0, and Advanced 2025 combined three solutions in a limit. Write the solution explicitly before taking any limit; do not guess from the equation.
Advanced 2024 gave limt→x (t10f(x) − x10f(t))/(t − x) = 0. The limit is the derivative of the numerator with respect to t at t = x, which gives 10x9f(x) − x10f′(x) = 0, so f(x) = 2x10 when f(1) = 2. Recognising a hidden derivative is the key skill.
Advanced 2019 described a curve only through its tangent: the tangent at P meets the y-axis at YP, and PYP = 1. Using the length of the tangent segment gives x√(1 + y′2) = 1, a separable equation. The tangent and normal formulas from Application of Derivatives are what turn such words into an equation.
"In Advanced, solving the equation gives you the function. The marks come from what you do with it afterwards: its maximum, its limit, its monotonicity. Practise the full question, not just the solving." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
The last step in how to study Differential Equations for JEE is an honest self-check. You should be able to answer "yes" to every line below without notes.
Differential Equations gave JEE Main 79 questions from 2024 to 2026 and JEE Advanced 15 questions from 2015 to 2026. Linear equations alone solved 39 of the Main questions, and with variables separable and homogeneous equations the three methods covered 60 of 79. Integral and functional equations are the one group that is growing. So if you are working out how to study Differential Equations for JEE, the order is clear: master the integrating-factor routine first, then separable equations and their substitutions, then integral and functional equations, then homogeneous equations, and keep exact forms and applications as quick wins.
How we built this data. We went through every JEE Main shift paper from 2024, 2025 and 2026 and every JEE Advanced paper from 2015 to 2026, picked out each question from this chapter and tagged it by the method that solves it. Where the archived text of a question was too incomplete to classify with confidence, we counted it in the totals and said so instead of guessing. One Advanced question filed here from another chapter is left out, as the footnote explains.
If you want a faculty member to check your preparation, every pack starts with a free 30-minute mentorship call. See all Maths chapter packs on our JEE Maths coaching page, and once your exam is done, our JEE counselling team can help you turn your rank into the right college and branch.
JEE Main asked 37 questions from this chapter in 2024 (6.17% of Maths questions), 21 in 2025 (4.42%) and 21 in 2026 (4.42%), counting all shifts. That is about one question per paper, often two in 2024.
JEE Main questions are mostly medium. Once you recognise the type, the method is fixed, and the difficulty is in the integration and the substitution. JEE Advanced questions are harder because they ask multi-correct statements about the solution. If you follow this guide on how to study Differential Equations for JEE, the Main questions become routine with steady practice.
Yes. It was asked 79 times in JEE Main from 2024 to 2026 and in 10 of the 12 JEE Advanced exams from 2015 to 2026. It also tests your integration, so preparing it strengthens the whole of Calculus.
No. Skipping it gives up about one question per JEE Main paper. Because half the questions use one method, the integrating factor, it is one of the quickest Maths chapters to turn into reliable marks.
Master linear equations first, then variables separable, then homogeneous equations. In JEE Main 2024–2026 these three methods solved 60 of the 79 questions, about three-quarters of the chapter. Add integral and functional equations and you cover 69 of 79.
JEE Advanced uses the same methods (linear equations led with 6 of 15 questions from 2015 to 2026) but asks more about the solution afterwards: its maximum, limit, periodicity or monotonicity. It also gives conditions as limits or integrals and describes curves through their tangents. 6 of the 15 Advanced questions were multi-correct.
