Module 2 · GATE Engineering Mathematics

Differential Equations

First and second order ODEs, Laplace transforms, applications.

First Order ODE · Second Order ODE · Laplace Transform

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Learning Objectives

Understand the fundamental concepts of Differential Equations
Apply key formulas and techniques to solve problems
Practice with exam-level questions to build speed and accuracy

Key Concepts

What are Differential Equations?

Differential equations (DEs) describe relationships involving rates of change and are fundamental to engineering analysis. GATE covers first and second order ordinary differential equations (ODEs), methods of solution, and Laplace transforms. Weightage is typically 4-6% in GATE.

Key Concepts

1. First Order ODE: Variable separable: dy/dx = f(x)g(y) → ∫dy/g(y) = ∫f(x)dx. Homogeneous: dy/dx = f(y/x) → substitute y = vx. Linear: dy/dx + P(x)y = Q(x) → integrating factor IF = e^(∫P dx) → solution: y(IF) = ∫Q(IF) dx. Bernoulli: dy/dx + Py = Qyⁿ → substitute v = y^(1-n).

2. Exact Differential Equations: M dx + N dy = 0 is exact if ∂M/∂y = ∂N/∂x. Solution: ∫M dx + ∫(terms in N not containing x) dy = C. If not exact, find integrating factor.

3. Second Order Linear ODE: a d²y/dx² + b dy/dx + cy = f(x). Homogeneous (f=0): auxiliary equation ar² + br + c = 0. Roots: real distinct → y = C₁e^(r₁x) + C₂e^(r₂x). Real equal → y = (C₁ + C₂x)e^(rx). Complex → y = e^(αx)(C₁cosβx + C₂sinβx). Particular integral for f(x) using method of undetermined coefficients or variation of parameters.

4. Laplace Transforms: L{f(t)} = ∫(0 to ∞) e⁻ˢᵗ f(t) dt. Standard transforms: L{1} = 1/s, L{tⁿ} = n!/sⁿ⁺¹, L{e^(at)} = 1/(s-a), L{sinωt} = ω/(s²+ω²), L{cosωt} = s/(s²+ω²). Properties: linearity, first shifting theorem L{e^(at)f(t)} = F(s-a), second shifting theorem L{f(t-a)H(t-a)} = e⁻ᵃˢF(s). Inverse Laplace: partial fractions + standard forms.

Solved Examples

Example 1 (First Order Linear ODE)

Solve: dy/dx + 2xy = 2x, given y(0) = 1.

a) y = 1 b) y = 1 + e^(-x²) c) y = 1 - e^(-x²) d) y = e^(-x²)

Solution: a) y = 1. P(x) = 2x, Q(x) = 2x. IF = e^(∫2x dx) = e^(x²). Solution: y·e^(x²) = ∫2x·e^(x²) dx = e^(x²) + C. So y = 1 + Ce^(-x²). Using y(0)=1: 1 = 1 + C → C=0. Therefore y = 1.

Example 2 (Second Order ODE)

Solve: d²y/dx² + 4y = 0, y(0)=1, y'(0)=2.

a) y = cos2x + sin2x b) y = cos2x + 2sin2x c) y = 2cos2x + sin2x d) y = cos2x + 3sin2x

Solution: a) y = cos2x + sin2x. AE: r² + 4 = 0 → r = ±2i. Solution: y = C₁cos2x + C₂sin2x. y(0)=1: C₁=1. y' = -2C₁sin2x + 2C₂cos2x. y'(0)=2: 2C₂=2 → C₂=1. So y = cos2x + sin2x.

Example 3 (Laplace Transform)

Find the Laplace transform of f(t) = 3 + 2t - e^(-3t).

a) 3/s + 2/s² - 1/(s+3) b) 3/s + 2/s² + 1/(s+3) c) 3/s - 2/s² - 1/(s-3) d) 1/s + 2/s² - 3/(s+3)

Solution: a) 3/s + 2/s² - 1/(s+3). L{3} = 3/s. L{2t} = 2·1/s² = 2/s². L{e^(-3t)} = 1/(s+3). By linearity: L{f(t)} = 3/s + 2/s² - 1/(s+3).

Shortcut Techniques

For variable separable, integrate both sides directly — no special method needed. For linear DE, always find IF first.
In second order ODE with constant coefficients, memorise the three cases of auxiliary equation roots.
For Laplace transforms, memorise the standard transform pairs and use partial fractions for inverse transforms.
Unit step function H(t-a) shifts the function: L{f(t-a)H(t-a)} = e^(-as)F(s).
Initial and final value theorems: lim(t→0) f(t) = lim(s→∞) sF(s), lim(t→∞) f(t) = lim(s→0) sF(s).

Pro Tip

In GATE, differential equations questions are highly formula-driven. Practice identifying the type of DE quickly — this determines the solution method. For Laplace transforms, focus on solving initial value problems by transforming the differential equation into an algebraic equation in s, which is then solved and inverse-transformed.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Practice Questions

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