JEE Mathematics: Differential Equations - Exact DE, Bernoulli's Equation, Applications

Differential Equations — Exact DE, Bernoulli's Equation, Applications

What This Is and Why It Matters for JEE

Differential Equations (DE) are crucial for physics, chemistry, and mathematics. In JEE, you can expect 2-3 questions every year, mostly on exact DE and Bernoulli's Equation. These questions can be moderate to tough, and it's essential for both Main and Advanced.

Prerequisites

• Calculus: Familiarity with functions, derivatives, and integrals.
• Algebra: Understanding of equations, inequalities, and functions.
• Mathematical Analysis: Basic knowledge of limits, continuity, and differentiability.

Quick Revision Path

If you're not confident in these topics, revise them quickly using online resources or your textbook.

Core Concepts (Exam-Focused)

• Exact DE: A DE of the form M(x, y)dx + N(x, y)dy = 0 is exact if ?M/?y = ?N/?x.
• Bernoulli's Equation: A DE of the form dy/dx + P(x)y = Q(x) can be solved using substitution y = u/v.
• Unit Conventions: Use dy/dx for derivatives and ? for integrals.

Step-by-Step Problem-Solving Strategy

1. Identify the type of DE (exact, Bernoulli's, or otherwise).
2. Check if the DE is exact by verifying ?M/?y = ?N/?x.
3. If exact, find the potential function f(x, y) using ?f/?x = M and ?f/?y = N.
4. Avoid trying to integrate M(x, y)dx + N(x, y)dy = 0 directly without checking for exactness.

Important Graphs / Diagrams (if applicable)

No specific graphs are required for this topic, but you should be able to sketch the direction field of a DE.

Typical JEE Question Patterns

• Find the general solution of a DE.
• Compare the time periods of two DEs.
• Determine the stability of a DE.

Common Mistakes & Exam Traps

• The mistake: Assuming a DE is exact without verifying ?M/?y = ?N/?x.
• Why it happens: Rushing through the problem or misreading the DE.
• How to avoid it: Verify the condition ?M/?y = ?N/?x before proceeding.

• Exam board insight: Examiners penalise incorrect assumptions.

• The mistake: Using the wrong substitution for Bernoulli's Equation.

• Why it happens: Misunderstanding the equation or substituting incorrectly.
• How to avoid it: Use y = u/v and substitute correctly.
• Exam board insight: Examiners expect correct substitution.

Time-Saving Shortcuts

• Use the test for exactness to quickly determine if a DE is exact.
• Verify the potential function using ?f/?x = M and ?f/?y = N.

Practice MCQs (Exam-Style)

Question 1: (Easy) Find the general solution of the DE dy/dx = 2x. A) y = x^2 + C B) y = x^2 - C C) y = x^2 + 2C D) y = x^2 - 2C

Answer: A) y = x^2 + C Solution: Integrate dy/dx = 2x to get y = x^2 + C. Common Wrong Answer: B) y = x^2 - C, which assumes dy/dx = -2x.

Question 2: (Moderate) Compare the time periods of two DEs dy/dx = -2y and dy/dx = -4y. A) The time periods are equal. B) The time period of the first DE is twice that of the second. C) The time period of the second DE is twice that of the first. D) The time periods are not comparable.

Answer: B) The time period of the first DE is twice that of the second. Solution: Use the formula T = 1/? to compare the time periods. Common Wrong Answer: A) The time periods are equal, which assumes ? is the same for both DEs.

Question 3: (JEE Advanced level) Determine the stability of the DE dy/dx = y^2 - 4. A) The equilibrium point is stable. B) The equilibrium point is unstable. C) The equilibrium point is neutral. D) The stability cannot be determined.

Answer: B) The equilibrium point is unstable. Solution: Analyze the DE to determine the stability of the equilibrium point. Common Wrong Answer: A) The equilibrium point is stable, which assumes dy/dx = -y^2 + 4.

Quick Revision Card (60-Second Summary)

• Exact DE: M(x, y)dx + N(x, y)dy = 0 is exact if ?M/?y = ?N/?x.
• Bernoulli's Equation: Use substitution y = u/v to solve.
• Potential function: Verify ?f/?x = M and ?f/?y = N.
• Direction field: Sketch the direction field of a DE.
• Time period: Compare time periods using T = 1/?.
• Stability: Analyze the DE to determine stability.

If You Get Stuck in Exam

• Write partial marks: If unsure, write what you know and get partial marks.
• Eliminate distractors: Eliminate options that are clearly incorrect.
• Skip and return: If stuck, skip the question and return to it later.

Related JEE Topics

• Separable DE: A DE that can be separated into two functions.
• Linear DE: A DE of the form dy/dx + P(x)y = Q(x).
• Homogeneous DE: A DE of the form dy/dx = f(y/x).