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Differential Equations

DE full marks: rewrite standard form → stamp SEP / HOM / LIN → apply only that recipe → use IC → optionally differentiate to verify.

Portion CalculusFocus Medium–High — classification decides everythingTarget 60–90sPortion share 28.5%Trend Falling share
JEE Advanced focuses on first-order: variable separable, homogeneous, linear (IF method). Misclassification is the main failure mode — not integration skill. DE scoring is classification first; every wrong template starts with a wrong stamp.

How Advanced asks this

  • Solve separable / homogeneous / linear DE
  • Particular solution with initial condition
  • Matching DE ↔ general solution
  • MC on which method applies
  • Order/degree identification

Traps that leak marks

  • Wrong class stamp
  • Missing product rule for y=vx
  • Dropping absolute value in ln
  • Using IF on non-linear forms
  • Treating homogeneous as separable without the v-sub
  • Wrong IF exponent (forgetting ∫P or the sign of P)
  • Applying IC before integrating both sides cleanly

01. Separable: dy/g(y)=f(x)dx. Homogeneous: dy/dx=f(y/x) → y=vx. Linear: dy/dx+P(x)y=Q(x) → IF=e^{∫P dx}.

02. Formation of DE / order-degree ideas appear; keep order = highest derivative order, degree = power of that derivative after clearing radicals.

03. After solving, a 10s differentiate-and-plug check catches algebra slips when time allows.

04. Keep ln|·| unless domain guarantees sign.

05. Advanced lane: classify in 15s — separable / homogeneous / linear / exact-looking. Misclassification is the mark killer; the solving template only works after the stamp.

06. Linear first-order: IF = e^{∫P dx}; multiply through and recognise d/dx[y·IF]. Initial conditions apply only after the general solution is clean.

Official · 2026

Syllabus for this topic

Official syllabus points for Differential Equations only — not the full Calculus portion map. Portion hub: Calculus · Calculus analysis.

Syllabus on this page is topic-scoped: Differential equations. The parent portion (Calculus) also covers other topics — use the portion hub for the full map.

Must cover (this topic)

  • Homogeneous, separable, and linear first-order DEs

Gaps / watch-outs

No topic-specific exclusions flagged — finish the unit points below for DE.

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Differential equations

high

Exam focusSeparable, homogeneous, and linear first-order DEs as Advanced asks them.

Syllabus points

  • Formation of ODEs; homogeneous first-order; separation of variables; linear first-order DE

Full Mathematics syllabus →

Cross-topic stems

Mixed joins for this topic

Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Differential Equations — practise the knowledge bridge, not only the single-chapter drill.

Mathematics2026 · Paper 2 · Q15–Q16linked-numerical

Exponential curves + area

Calculus – Curves+Definite Integration – Area+Trigonometric equations

Knowledge bridge

Step 1: set y = e^{-x} equal to y = e^{-x}(sin x + cos x) and cancel the common exponential to get a trig equation for intersection abscissae α_i. Step 2: order those roots and set up the definite integral of the difference of the two curves between α_1 and α_4 for the enclosed area β. Step 3: simplify the given log expression in β to a clean numerical value.

Why it feels hard

The stem hides trig solving inside an exponential envelope; students often integrate before finding exact limits.

What to practise

Drill ‘cancel common positive factor → trig roots → area integral’ as one reflex chain.

2026 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2026 · Paper 2 · Q8multi-correct

DE solution + extrema + intersections

Differential Equations+Functions / AOD+Curve intersections

Knowledge bridge

Step 1: solve x y' = y − x³ (homogeneous/linear in y/x) to get the family y = Cx − x³/2. Step 2: impose the given condition to fix C, then use y' = 0 to locate the local max. Step 3: compare monotonicity on (1,2) and count positive roots of f = g.

Why it feels hard

Multi-correct options test DE, calculus and graph reading together; one sign error in the DE kills several options.

What to practise

After every DE, immediately ask: extrema? monotonicity? intersections with a given g?

2026 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2025 · Paper 2 · Q1–Q2multi-concept

Limit/derivative structure + area inequalities

Calculus – Limits / AOD+Definite Integration – Area+Inequalities of regions

Knowledge bridge

Step 1: analyse the given limit expression to identify the correct f(x) / derivative behaviour. Step 2: on a related (or following) stem, sketch the region defined by inequalities. Step 3: integrate carefully (logs appear from 1/x-type boundaries).

Why it feels hard

Back-to-back calculus depth: analytic limit then geometric area.

What to practise

Pair every limit/AOD drill with one area-of-region numerical the same day.

2025 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2025 · Paper 1 · Q13numerical

Parabola numerical with algebraic constraints

Coordinate Geometry – Parabola+Algebraic parameters+Numerical answer

Knowledge bridge

Step 1: write the parabola in standard form and express the geometric condition algebraically. Step 2: reduce to an equation in the unknown parameter. Step 3: extract the required integer/decimal for the NUM grid.

Why it feels hard

Conic geometry collapses into algebra; diagram alone is not enough.

What to practise

Translate every parabola condition into an equation before computing.

2025 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2024 · Paper 1 · Q4single-correct

Ellipse single-correct with parametric depth

Ellipse+Tangents / chords+Algebraic conditions

Knowledge bridge

Step 1: place the ellipse in standard form and introduce eccentric angle or slope form. Step 2: impose the geometric condition (tangent/chord/focus). Step 3: match the simplified relation to the correct option.

Why it feels hard

Parametric slips create attractive wrong options.

What to practise

Keep both slope and parametric forms of ellipse tangents fluent.

2024 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2021 · Paper 1 · Q8–Q10linked-numerical

Straight-line linked numerical stem

Coordinate Geometry – Straight line+Linked numerical stems+Distance / intercept algebra

Knowledge bridge

Step 1: extract all shared geometric data from the paragraph once. Step 2: answer the first NUM (slope/intercept/distance). Step 3: reuse intermediates for the linked follow-ups without re-deriving from scratch.

Why it feels hard

Early example of Advanced’s linked-stem tax on organisation.

What to practise

Box shared quantities at the top of linked stems — treat them as a mini formula sheet.

2021 year review →Mathematics notes →Quick card →Syllabus →

All mixed joins →

Why methods matter

Common methods that change your score

DE full marks = classification. These methods stamp SEP/HOM/LIN before any integration — misclassification is what actually loses marks, not hard integrals.

01

Classify first

Misclassification is the real DE killer — this stamp fixes the root cause.

Do thisOne-word stamp before any integration.

Why it works

Stops mixing IF with y=vx mid-way.

See it once

dy/dx+(2/x)y=x → LIN, IF=x², not homogeneous guesswork.

02

IF mechanics

Linear DE becomes one product-rule recognition, not a mystery.

Do thisWrite IF, multiply through, recognise derivative of product.

Why it works

Linear DE becomes one quadrature.

See it once

d/dx(x²y)=x³ ⇒ x²y=x⁴/4+C.

03

Homogeneous product rule

Catches the algebra leak that makes y=vx solutions wrong.

Do thisWhen y=vx, differentiate with product rule: v+x v′.

Why it works

Common algebra leak.

See it once

Forget product rule ⇒ wrong separable equation in v.

04

IC last

Keeps general solution clean so particular solution algebra does not tangle early.

Do thisFind general solution, then plug IC for C.

Why it works

Keeps algebra organised.

See it once

Don’t insert IC before integrating both sides completely.

Processed examples

Sample questions · full working

Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.

01

standardFamily · Linear first-order

Sample question

Solve dy/dx + (2/x)y = x (x>0).

Classification LIN → IF is the whole speed story.

Model chosenStamp LIN; IF=x².

Processed solution

  1. P=2/x, Q=x ⇒ IF=e^{∫2/x dx}=x².
  2. d/dx(x²y)=x³ ⇒ x²y = x⁴/4 + C.
  3. y = x²/4 + C/x².

Answery = x²/4 + C/x²

Takeaway: IF recipe only after LIN stamp.

02

bankFamily · Separable

Sample question

Solve dy/dx = y (y>0) with y(0)=2.

Separable + IC is the cleanest DE bank question — still stamp SEP first.

Model chosenSEP: dy/y = dx.

Processed solution

  1. ∫ dy/y = ∫ dx ⇒ ln y = x + K.
  2. y = Ae^x; y(0)=2 ⇒ A=2.
  3. y=2e^x.

Answery = 2eˣ

Takeaway: IC after general solution.

03

twistFamily · Homogeneous first-order

Sample question

Solve dy/dx = (x + y)/(x − y) with y(1) = 0, giving the solution implicitly for x > 0.

Looks linear-ish but is homogeneous — misclassification is the mark killer.

Model chosenStamp HOM → y = vx → separate → apply IC.

Processed solution

  1. Degree match ⇒ homogeneous. Set y = vx ⇒ v + x v′ = (1 + v)/(1 − v).
  2. Then x v′ = (1 + v)/(1 − v) − v = (1 + v²)/(1 − v).
  3. Separate: (1 − v)/(1 + v²) dv = dx/x.
  4. Integrate: arctan v − (1/2) ln(1 + v²) = ln x + C.
  5. IC (1, 0) ⇒ v = 0 ⇒ C = 0. Replace v = y/x.

Answertan⁻¹(y/x) − (1/2) ln(1 + y²/x²) = ln x

Takeaway: Equal-degree rational DEs → try v = y/x immediately.

Reusable models

Common question families

Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.

01

Linear first-order

Looks likedy/dx + P(x)y = Q(x)

Fast modelStamp LIN → IF=e^{∫P} → d/dx(y·IF)=Q·IF.

Same solve path

  1. Identify P,Q.
  2. IF.
  3. Integrate Q·IF.
  4. Divide by IF; apply IC.

Rough skeleton

  • P,Q
  • IF
  • ∫
  • y

Why faster

One recipe — no method mixing.

Refuse this

Do not try y=vx on a clearly linear DE.

02

Homogeneous first-order

Looks likedy/dx = f(y/x) or degree-balanced M,N

Fast modelStamp HOM → y=vx → remember product rule v+x v′.

Same solve path

  1. Confirm homogeneous.
  2. y=vx substitute.
  3. Separate v,x.
  4. Back-substitute; IC if any.

Rough skeleton

  • HOM
  • v
  • Integrate
  • y

Why faster

Classification redirects you away from illegal IF.

Refuse this

Do not forget product rule when differentiating y=vx.

03

Separable

Looks likeg(y) dy = f(x) dx after rearrange

Fast modelStamp SEP → split → integrate both sides → +C → IC.

Same solve path

  1. Separate variables.
  2. Integrate.
  3. Implicit/explicit solve if asked.
  4. Apply IC.

Rough skeleton

  • SEP
  • ∫
  • C
  • Particular

Why faster

Shortest DE family when recognised early.

Refuse this

Do not force IF when variables already separate.

04

Matching DE ↔ solution

Looks likeList of DEs vs general solutions

Fast modelDifferentiate the proposed solution (or plug) independently for each list item.

Same solve path

  1. For each solution, differentiate.
  2. See which DE it satisfies.
  3. Independent matches — no code forcing.

Rough skeleton

  • Item
  • y′
  • Matches DE?

Why faster

Verify-by-differentiating is faster than re-solving each DE.

Refuse this

Do not solve all four DEs from scratch if plugging works.

Hall timing

60–90s process

  1. 0–15s
    Rewrite dy/dx form. Note if IC given.

    Clear differentials into standard first-order shape.

  2. 15–35s
    Stamp SEP / HOM / LIN (one word on rough sheet).

    If M(x,y)dx+N dy=0, check homogeneous degree or exactness only if in syllabus path; Advanced usually redirects to these three.

  3. 35–75s
    Apply matching recipe only. Integrate carefully; apply IC.

    Linear: multiply by IF, left side becomes d/dx(y·IF).

  4. 75–90s
    Optional verify by differentiating. Check |r|-style domain issues for ln.

    Constants fixed by IC must satisfy original point.

After the model is locked

Quick formula bank

Separable

∫ dy/g(y) = ∫ f(x) dx

Variables separate cleanly

Homogeneous

y=vx ⇒ dy/dx = v+x dv/dx = f(v)

f(y/x) form

Linear IF

IF=e^{∫P dx}; d/dx(y·IF)=Q·IF

dy/dx+Py=Q

Order / degree

Order = highest derivative order; degree = power of that derivative

Classification MC

General solution

Contains arbitrary constants equal in number to order (typical first-order → one C)

Matching solution forms

Mental bank

  • Stamp SEP/HOM/LIN first
  • IF=e^{∫P dx}
  • Homogeneous: y=vx
  • Separable: split and integrate
  • Keep ln|·|
  • IC after general solution
  • Separable: dy/dx=g(x)h(y) → ∫dy/h=∫g dx
  • Homogeneous: y=vx (or x=uy) after degree check
  • Linear: dy/dx+Py=Q → IF=e^{∫P dx}
  • IC plugged last into general solution + C

Make it automatic

Practice · checklist

Daily

  • 1 DE rotating SEP/HOM/LIN.
  • Write class stamp by 15s every time.
  • 1 IC particular solution (alt days).

Weekly

  • 1 matching DE↔solution.
  • 1 mixed set of 8 first-order DEs.
  • Error log: misclassification.

100% checklist

  • Every DE starts with class stamp.
  • IF/y=vx/separable used correctly.
  • IC applied after general solution.
  • Optional verify when time allows.
  • ≥90% on last timed DE set.
  • Equation type stamped before any substitution.
  • IF written explicitly for every linear first-order.
  • Particular solution only after general +C is ready.
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