Differential Equations (ODEs & PDEs)

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📘 Comprehensive Syllabus & Examination Guide

Differential Equations (ODEs & PDEs)

Official curriculum roadmap, subject/topic distribution, negative marking rules, pacing guidelines, and solved sample questions.

🎯 Mapped Subjects & Topic Question Distribution

Total Question Pool 100%
100 MCQs
Combined Active Syllabus
Differential Equations (ODEs & PDEs)
100 MCQs
Topic Pool
📊 Question Pool Structure
100 MCQs across fundamental, intermediate, and advanced concept tiers.
⚡ Recommended Pacing
45 to 60 seconds per MCQ. Flag complex problems and preserve 10 minutes for final revision.
⚖️ Scoring & Negative Marking
+1 mark per correct answer. In competitive tests with negative marking, -0.25 applies for incorrect guesses.

💡 Strategic Preparation & Exam Hall Guidelines

To maximize your score on Differential Equations (ODEs & PDEs), candidates are advised to follow a structured three-pass approach. In the First Pass, solve all direct recall and formula-based questions within 30 seconds each to secure foundational marks. In the Second Pass, tackle multi-step analytical and quantitative reasoning problems. In the Third Pass, review marked questions and verify calculations.

Practice with the interactive player below to evaluate your speed and accuracy under real exam pressure. Every question features full mathematical formulas, step-by-step worked solutions, and conceptual explanations vetted by Apex Rankers Academy subject matter specialists.

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Solved Blueprint Examples

📝 Pre-Rendered Solved Sample Questions & Detailed Solutions

Showing 10 solved representative questions

Review the solved problems below to understand question phrasing, answer choices, and step-by-step solution logic prior to starting the full interactive practice drill:

Sample Question 1
Differential Equations (ODEs & PDEs) MEDIUM • MULTIPLE_CHOICE
The integrating factor (IF) for the linear first-order differential equation dy/dx + P(x)y = Q(x) is:
A ln(P(x))
B e^(∫ P(x) dx)
C ∫ P(x) dx
D e^(P(x))
✓ Correct Answer: B - e^(∫ P(x) dx)
📖 Step-by-Step Solution & Conceptual Rationale:
Multiplying by e^(∫ P dx) turns the left side into exact derivative d/dx [y e^(∫ P dx)].
Sample Question 2
Differential Equations (ODEs & PDEs) MEDIUM • MULTIPLE_CHOICE
The general solution to the homogeneous ODE y'' + 4y = 0 is:
A y = c1 e^(2x) + c2 e^(-2x)
B y = c1 sin(4x)
C y = c1 cos(2x) + c2 sin(2x)
D y = c1 x² + c2
✓ Correct Answer: C - y = c1 cos(2x) + c2 sin(2x)
📖 Step-by-Step Solution & Conceptual Rationale:
Characteristic equation r² + 4 = 0 gives imaginary roots r = ±2i.
Sample Question 3
Differential Equations (ODEs & PDEs) EASY • MULTIPLE_CHOICE
The Laplace Transform of f(t) = e^(at) for s > a is:
A a / s
B s / (s² + a²)
C 1 / (s - a)
D 1 / (s + a)
✓ Correct Answer: C - 1 / (s - a)
📖 Step-by-Step Solution & Conceptual Rationale:
L{e^(at)} = ∫ e^(-st) e^(at) dt = ∫ e^(-(s-a)t) dt = 1/(s-a).
Sample Question 4
Differential Equations (ODEs & PDEs) HARD • MULTIPLE_CHOICE
The one-dimensional heat conduction equation ∂u/∂t = α ∂²u/∂x² is classified as:
A Hyperbolic PDE
B Nonlinear wave equation
C Parabolic partial differential equation
D Elliptic PDE
✓ Correct Answer: C - Parabolic partial differential equation
📖 Step-by-Step Solution & Conceptual Rationale:
Discriminant B² - 4AC = 0 classifies diffusion/heat equations as parabolic.
Sample Question 5
Differential Equations (ODEs & PDEs) HARD • MULTIPLE_CHOICE
The wave equation ∂²u/∂t² = c² ∂²u/∂x² is classified as:
A Elliptic PDE
B Hyperbolic partial differential equation
C Parabolic PDE
D Poisson equation
✓ Correct Answer: B - Hyperbolic partial differential equation
📖 Step-by-Step Solution & Conceptual Rationale:
B² - 4AC > 0 characterizes hyperbolic wave propagating equations.
Sample Question 6
Differential Equations (ODEs & PDEs) MEDIUM • MULTIPLE_CHOICE
In professional practice: The integrating factor (IF) for the linear first-order differential equation dy/dx + P(x)y = Q(x) is:
A ln(P(x))
B e^(P(x))
C ∫ P(x) dx
D e^(∫ P(x) dx)
✓ Correct Answer: D - e^(∫ P(x) dx)
📖 Step-by-Step Solution & Conceptual Rationale:
Multiplying by e^(∫ P dx) turns the left side into exact derivative d/dx [y e^(∫ P dx)].
Sample Question 7
Differential Equations (ODEs & PDEs) MEDIUM • MULTIPLE_CHOICE
In professional practice: The general solution to the homogeneous ODE y'' + 4y = 0 is:
A y = c1 e^(2x) + c2 e^(-2x)
B y = c1 x² + c2
C y = c1 sin(4x)
D y = c1 cos(2x) + c2 sin(2x)
✓ Correct Answer: D - y = c1 cos(2x) + c2 sin(2x)
📖 Step-by-Step Solution & Conceptual Rationale:
Characteristic equation r² + 4 = 0 gives imaginary roots r = ±2i.
Sample Question 8
Differential Equations (ODEs & PDEs) EASY • MULTIPLE_CHOICE
In professional practice: The Laplace Transform of f(t) = e^(at) for s > a is:
A s / (s² + a²)
B 1 / (s + a)
C 1 / (s - a)
D a / s
✓ Correct Answer: C - 1 / (s - a)
📖 Step-by-Step Solution & Conceptual Rationale:
L{e^(at)} = ∫ e^(-st) e^(at) dt = ∫ e^(-(s-a)t) dt = 1/(s-a).
Sample Question 9
Differential Equations (ODEs & PDEs) HARD • MULTIPLE_CHOICE
In professional practice: The one-dimensional heat conduction equation ∂u/∂t = α ∂²u/∂x² is classified as:
A Elliptic PDE
B Hyperbolic PDE
C Parabolic partial differential equation
D Nonlinear wave equation
✓ Correct Answer: C - Parabolic partial differential equation
📖 Step-by-Step Solution & Conceptual Rationale:
Discriminant B² - 4AC = 0 classifies diffusion/heat equations as parabolic.
Sample Question 10
Differential Equations (ODEs & PDEs) HARD • MULTIPLE_CHOICE
In professional practice: The wave equation ∂²u/∂t² = c² ∂²u/∂x² is classified as:
A Hyperbolic partial differential equation
B Elliptic PDE
C Poisson equation
D Parabolic PDE
✓ Correct Answer: A - Hyperbolic partial differential equation
📖 Step-by-Step Solution & Conceptual Rationale:
B² - 4AC > 0 characterizes hyperbolic wave propagating equations.
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