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Differential Equations (ODEs & PDEs) (Pure & Applied Mathematics) Solved Questions & Notes (2026) - Apex Rankers

Natural & Physical Sciences > Pure & Applied Mathematics > Differential Equations (ODEs & PDEs)

100 Total Questions
~150 mins Read Time
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Q. 1 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
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)
✓ Correct
C
∫ P(x) dx
D
e^(P(x))
💡 Step-by-Step Explanation & Concept Rationale
Multiplying by e^(∫ P dx) turns the left side into exact derivative d/dx [y e^(∫ P dx)].
Q. 2 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
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)
✓ Correct
D
y = c1 x² + c2
💡 Step-by-Step Explanation & Concept Rationale
Characteristic equation r² + 4 = 0 gives imaginary roots r = ±2i.
Q. 3 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
The Laplace Transform of f(t) = e^(at) for s > a is:
A
a / s
B
s / (s² + a²)
C
1 / (s - a)
✓ Correct
D
1 / (s + a)
💡 Step-by-Step Explanation & Concept Rationale
L{e^(at)} = ∫ e^(-st) e^(at) dt = ∫ e^(-(s-a)t) dt = 1/(s-a).
Q. 4 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
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
✓ Correct
D
Elliptic PDE
💡 Step-by-Step Explanation & Concept Rationale
Discriminant B² - 4AC = 0 classifies diffusion/heat equations as parabolic.
Q. 5 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
The wave equation ∂²u/∂t² = c² ∂²u/∂x² is classified as:
A
Elliptic PDE
B
Hyperbolic partial differential equation
✓ Correct
C
Parabolic PDE
D
Poisson equation
💡 Step-by-Step Explanation & Concept Rationale
B² - 4AC > 0 characterizes hyperbolic wave propagating equations.
Q. 6 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
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
💡 Step-by-Step Explanation & Concept Rationale
Multiplying by e^(∫ P dx) turns the left side into exact derivative d/dx [y e^(∫ P dx)].
Q. 7 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
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
💡 Step-by-Step Explanation & Concept Rationale
Characteristic equation r² + 4 = 0 gives imaginary roots r = ±2i.
Q. 8 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
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)
✓ Correct
D
a / s
💡 Step-by-Step Explanation & Concept Rationale
L{e^(at)} = ∫ e^(-st) e^(at) dt = ∫ e^(-(s-a)t) dt = 1/(s-a).
Q. 9 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
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
✓ Correct
D
Nonlinear wave equation
💡 Step-by-Step Explanation & Concept Rationale
Discriminant B² - 4AC = 0 classifies diffusion/heat equations as parabolic.
Q. 10 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
In professional practice: The wave equation ∂²u/∂t² = c² ∂²u/∂x² is classified as:
A
Hyperbolic partial differential equation
✓ Correct
B
Elliptic PDE
C
Poisson equation
D
Parabolic PDE
💡 Step-by-Step Explanation & Concept Rationale
B² - 4AC > 0 characterizes hyperbolic wave propagating equations.
Q. 11 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: 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)
✓ Correct
C
∫ P(x) dx
D
e^(P(x))
💡 Step-by-Step Explanation & Concept Rationale
Multiplying by e^(∫ P dx) turns the left side into exact derivative d/dx [y e^(∫ P dx)].
Q. 12 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: The general solution to the homogeneous ODE y'' + 4y = 0 is:
A
y = c1 sin(4x)
B
y = c1 x² + c2
C
y = c1 e^(2x) + c2 e^(-2x)
D
y = c1 cos(2x) + c2 sin(2x)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Characteristic equation r² + 4 = 0 gives imaginary roots r = ±2i.
Q. 13 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: The Laplace Transform of f(t) = e^(at) for s > a is:
A
s / (s² + a²)
B
a / s
C
1 / (s + a)
D
1 / (s - a)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
L{e^(at)} = ∫ e^(-st) e^(at) dt = ∫ e^(-(s-a)t) dt = 1/(s-a).
Q. 14 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
According to standard examination standards: The one-dimensional heat conduction equation ∂u/∂t = α ∂²u/∂x² is classified as:
A
Elliptic PDE
B
Nonlinear wave equation
C
Parabolic partial differential equation
✓ Correct
D
Hyperbolic PDE
💡 Step-by-Step Explanation & Concept Rationale
Discriminant B² - 4AC = 0 classifies diffusion/heat equations as parabolic.
Q. 15 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
According to standard examination standards: The wave equation ∂²u/∂t² = c² ∂²u/∂x² is classified as:
A
Hyperbolic partial differential equation
✓ Correct
B
Elliptic PDE
C
Parabolic PDE
D
Poisson equation
💡 Step-by-Step Explanation & Concept Rationale
B² - 4AC > 0 characterizes hyperbolic wave propagating equations.
Q. 16 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: The integrating factor (IF) for the linear first-order differential equation dy/dx + P(x)y = Q(x) is:
A
e^(P(x))
B
e^(∫ P(x) dx)
✓ Correct
C
ln(P(x))
D
∫ P(x) dx
💡 Step-by-Step Explanation & Concept Rationale
Multiplying by e^(∫ P dx) turns the left side into exact derivative d/dx [y e^(∫ P dx)].
Q. 17 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: 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 cos(2x) + c2 sin(2x)
✓ Correct
D
y = c1 sin(4x)
💡 Step-by-Step Explanation & Concept Rationale
Characteristic equation r² + 4 = 0 gives imaginary roots r = ±2i.
Q. 18 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: The Laplace Transform of f(t) = e^(at) for s > a is:
A
a / s
B
1 / (s + a)
C
s / (s² + a²)
D
1 / (s - a)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
L{e^(at)} = ∫ e^(-st) e^(at) dt = ∫ e^(-(s-a)t) dt = 1/(s-a).
Q. 19 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
From an applied perspective: The one-dimensional heat conduction equation ∂u/∂t = α ∂²u/∂x² is classified as:
A
Elliptic PDE
B
Nonlinear wave equation
C
Hyperbolic PDE
D
Parabolic partial differential equation
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Discriminant B² - 4AC = 0 classifies diffusion/heat equations as parabolic.
Q. 20 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
From an applied perspective: The wave equation ∂²u/∂t² = c² ∂²u/∂x² is classified as:
A
Parabolic PDE
B
Elliptic PDE
C
Hyperbolic partial differential equation
✓ Correct
D
Poisson equation
💡 Step-by-Step Explanation & Concept Rationale
B² - 4AC > 0 characterizes hyperbolic wave propagating equations.
Q. 21 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: The integrating factor (IF) for the linear first-order differential equation dy/dx + P(x)y = Q(x) is:
A
e^(P(x))
B
∫ P(x) dx
C
ln(P(x))
D
e^(∫ P(x) dx)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Multiplying by e^(∫ P dx) turns the left side into exact derivative d/dx [y e^(∫ P dx)].
Q. 22 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: The general solution to the homogeneous ODE y'' + 4y = 0 is:
A
y = c1 x² + c2
B
y = c1 e^(2x) + c2 e^(-2x)
C
y = c1 cos(2x) + c2 sin(2x)
✓ Correct
D
y = c1 sin(4x)
💡 Step-by-Step Explanation & Concept Rationale
Characteristic equation r² + 4 = 0 gives imaginary roots r = ±2i.
Q. 23 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: The Laplace Transform of f(t) = e^(at) for s > a is:
A
s / (s² + a²)
B
1 / (s - a)
✓ Correct
C
1 / (s + a)
D
a / s
💡 Step-by-Step Explanation & Concept Rationale
L{e^(at)} = ∫ e^(-st) e^(at) dt = ∫ e^(-(s-a)t) dt = 1/(s-a).
Q. 24 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Under standard operational protocols: The one-dimensional heat conduction equation ∂u/∂t = α ∂²u/∂x² is classified as:
A
Parabolic partial differential equation
✓ Correct
B
Nonlinear wave equation
C
Elliptic PDE
D
Hyperbolic PDE
💡 Step-by-Step Explanation & Concept Rationale
Discriminant B² - 4AC = 0 classifies diffusion/heat equations as parabolic.
Q. 25 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Under standard operational protocols: The wave equation ∂²u/∂t² = c² ∂²u/∂x² is classified as:
A
Poisson equation
B
Hyperbolic partial differential equation
✓ Correct
C
Parabolic PDE
D
Elliptic PDE
💡 Step-by-Step Explanation & Concept Rationale
B² - 4AC > 0 characterizes hyperbolic wave propagating equations.
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