Differential Equations

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Differential Equations — Key Facts for JEE Main Differential equation: equation involving derivative of unknown function Order: highest derivative present; Degree: highest power of highest order derivative (when expressed as polynomial) General solution: family of curves; Particular solution: specific curve satisfying given condition First order linear: dy/dx + Py = Q; solution uses integrating factor IF = e^{∫P dx} ⚡ Exam tip: JEE Main tests solving differential equations and word problems — always find the particular solution using initial condition!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Differential Equations — JEE Main Study Guide

Variable separable: dy/dx = f(x)·g(y) → separate variables: dy/g(y) = f(x)dx → integrate both sides General solution: ∫ dy/g(y) = ∫ f(x) dx + C

Homogeneous differential equation: dy/dx = f(y/x); substitute y = vx, dy/dx = v + x·dv/dx Result: v + x·dv/dx = f(v) → separate v and x

Linear first order: dy/dx + P(x)·y = Q(x) Integrating Factor (IF) = e^{∫P dx} Solution: y·IF = ∫ Q·IF dx + C

Exact differential equation: M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x Solution: ∫ M dx + ∫ (terms with y only from N) dy = C

Non-exact → make exact: Multiply by integrating factor (function of x or y) For IF(x): IF(x) = e^{∫(∂M/∂y − ∂N/∂x)/N dx} when this is function of x only

Order and degree: Order = highest order derivative Degree = highest power of highest order derivative (after removing fractions and radicals, polynomial in derivatives) Not defined if: non-integer powers, fractional powers of derivative, derivatives inside log/trig functions

Formation of DE: From family of curves y = f(x, c): eliminate constant(s) c by differentiating and using original equation

Word problems: Common patterns: population growth/decay (dP/dt = kP), Newton’s law of cooling (dT/dt = k(T−T₀)), mixing problems

• Key formula: Linear: y·IF = ∫ Q·IF dx + C where IF = e^{∫P dx}; Variable separable: ∫ dy/g(y) = ∫ f(x) dx + C
• Common trap: Don’t forget the constant of integration C — it’s what makes it the general solution
• Exam weight: 1–2 questions per year (4–8 marks); word problems are very common

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Differential Equations — Comprehensive JEE Main Notes

Clairaut’s equation: y = xp + f(p) where p = dy/dx Solution: y = cx + f(c) (general solution) plus singular solution from envelope

Orthogonal trajectories: For family F(x, y, c) = 0, find differential equation by eliminating c: dy/dx = f(x, y, c) For orthogonal trajectories, replace dy/dx by −dx/dy: orthogonal DE: dy/dx = 1/f(x, y, c) Solve to get family of curves orthogonal to original

Second order linear with constant coefficients: ay” + by’ + cy = 0 → Characteristic equation: ar² + br + c = 0

• Real distinct roots r₁, r₂: y = C₁e^{r₁x} + C₂e^{r₂x}
• Real equal roots r: y = (C₁ + C₂x)e^{rx}
• Complex roots α ± iβ: y = e^{αx}[C₁ cos βx + C₂ sin βx]

Non-homogeneous second order: ay” + by’ + cy = f(x) where f(x) is particular solution form:

• f(x) = polynomial of degree n: try polynomial of same degree
• f(x) = e^{mx}: try Ae^{mx} (if m is root, multiply by x; if double root, multiply by x²)
• f(x) = sin/cos mx: try A sin mx + B cos mx

Method of variation of parameters: For y” + Py’ + Qy = R(x), find two solutions y₁, y₂ of homogeneous equation, then particular solution using integrals

Solving using substitution:

• Bernoulli: dy/dx + P(x)y = Q(x)y^n → divide by y^n, set z = y^{1−n} → linear in z
• Cauchy-Euler: x²y” + axy’ + by = 0 → try y = x^m → get m equation

Differential equation of family of curves: For n-parameter family, eliminate n constants by differentiating n times

Physical applications:

• Radioactive decay: dN/dt = −λN → N = N₀e^{−λt}
• Population growth: dP/dt = kP → P = P₀e^{kt}
• Newton’s cooling: dT/dt = −k(T − T₀) → T = T₀ + (T₁−T₀)e^{−kt}
• Simple harmonic: d²x/dt² + ω²x = 0 → x = A sin(ωt + φ)
• Damped oscillation: d²x/dt² + 2k·dx/dt + ω²x = 0

Differential operators: D = d/dx; operator P(D) = a₀D^n + a₁D^{n−1} + … + a_n P(D)(e^{mx}) = P(m)e^{mx} P(D)(e^{mx}·V) = e^{mx}·P(D+m)V

Particular solution for repeated roots: If m is root of multiplicity s: try x^s · (A₀ + A₁x + … + A_{s−1}x^{s−1})e^{mx}

Reduction of order: If y₁ is known solution of second order linear homogeneous, second solution y₂ = y₁∫ e^{−∫P dx}/y₁² dx

Boundary value problems: Given conditions at two different points (not just one initial condition) For second order: need two conditions to determine both constants

• Remember: Variable separable: dy/dx = f(x)g(y) → dy/g(y) = f(x)dx; Linear: IF = e^{∫P dx}; Exact: ∂M/∂y = ∂N/∂x; Second order homogeneous: solve characteristic equation, roots determine form
• Previous years: “Solve dy/dx + 2y = e^{3x}” [2023]; “Find general solution of (1+x²)dy/dx + 2xy = 1” [2024]; “Solve d²y/dx² + 4dy/dx + 4y = 0” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Differential Equations: 1–2 questions per year, 4–8 marks
• Common patterns: solve first order DE, word problems, find particular solution, orthogonal trajectories
• Weight: medium frequency, high scoring

💡 Pro Tips

• Identify the type first: separable, homogeneous, linear, or exact
• For linear first order dy/dx + Py = Q: IF = e^{∫P dx} is the key step
• For exact equations, always verify ∂M/∂y = ∂N/∂x before proceeding
• In word problems, translate the rate condition directly into DE
• Don’t forget the constant of integration — the problem may ask for general or particular solution
• For initial value problems, substitute and solve for C before writing final answer

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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