Karnataka 2nd PUC Maths Chapter 9 Important Questions 2027

Karnataka 2nd PUC Maths Chapter 9 Important Questions 2027 covers Differential Equations, an important chapter in the Karnataka 2nd PUC Syllabus. The chapter covers questions on identifying the order and degree of differential equations, finding their solutions, and forming differential equations from given conditions. 

Students can practise questions on variable separable equations, homogeneous differential equations and linear differential equations. KCET-based questions from different years also provide practice with different types of Differential Equations Questions and Answers.

Karnataka 2nd PUC Differential Equations Important Questions

The 2nd PUC Differential Equations Important Questions cover key areas such as order and degree, solutions of differential equations, variable separable equations, homogeneous differential equations and linear differential equations.

These Differential Equations Questions and Answers include different levels of practice along with KCET-based questions from previous years. They help students revise important concepts and practise different types of questions from Chapter 9.

• Differential Equations — 1 Mark Questions

This section covers fundamental objective-type questions on differential equations, testing key concepts such as order, degree, and the number of arbitrary constants in a general solution. These one-mark questions are ideal for quick revision and exam practice.

1. The order of a differential equation is always
(A) Rational number
(B) Whole number
(C) Negative integer
(D) Positive integer
Answer: (D) Positive integer

2. The number of arbitrary constants in the general solution of a differential equation of fourth order is
(A) 4
(B) 3
(C) 2
(D) 1
Answer: (A) 4

3. The order of a differential equation whose general solution is y = A sin x + B cos x is
(A) 4
(B) 2
(C) 0
(D) 3
Answer: (B) 2

4. The solution of the differential equation x dy − y dx = 0 represents
(A) A rectangular hyperbola
(B) Parabola whose vertex is at origin
(C) Straight line passing through origin
(D) Circle whose centre is origin
Answer: (C) Straight line passing through origin

• Differential Equations — 2 Mark Questions

These two-mark questions build on the basics, focusing on the degree and order of differential equations, the number of arbitrary constants in particular solutions, and identifying differential equations from their general solutions. They're useful for strengthening conceptual clarity before exams.

1. The sum of the degree and order of the differential equation d/dx[(dy/dx)³] = 0 is
(A) 2
(B) 3
(C) 4
(D) 5
Answer: (B) 3

2. Find the number of arbitrary constants in the particular solution of a differential equation of third order
(A) 3
(B) 2
(C) 1
(D) 0
Answer: (D) 0

3. The equation y = mx + c is the general solution of
(A) x(dy/dx) = y
(B) y(dy/dx) = x
(C) d²y/dx² = 0
(D) d²x/dy² = 0
Answer: (C) d²y/dx² = 0

4. The number of solutions of dy/dx = (y+1)/(x−1), when y(1) = 2, is
(A) Three
(B) One
(C) Infinite
(D) Two
Answer: (B) One

• Differential Equations — 3/5 Mark Questions

These are more involved practice questions covering higher-order concepts such as forming and solving differential equations, finding integrating factors, and applying initial conditions to determine specific curves. They are grouped here by difficulty level (Easy, Average, Difficult) rather than strict mark allocation, and are well-suited for deeper practice.

1. The equation of the curve passing through the point (1, 1), such that the slope of the tangent at any point (x, y) is equal to the product of its coordinates, is
(A) 2 log x = y² − 1
(B) 2 log y = x² − 1
(C) 2 log x = y² + 1
(D) 2 log y = x² + 1
Answer: (B) 2 log y = x² − 1

2. The integrating factor of (1+x²)(dy/dx) + xy = 1 is
(A) 1 + x²
(B) ½ log(1+x²)
(C) x/(1+x²)
(D) √(1+x²)
Answer: (D) √(1+x²)

3. The general solution of dy/dx + y tan x = sec x is
(A) y sec x = tan x + c
(B) y tan x = sec x + c
(C) y cosec x = tan x + c
(D) x sec x = tan y + c
Answer: (A) y sec x = tan x + c

4. The integrating factor of dy/dx = x + xy is e^(kx²). Then k is
(A) ½
(B) −½
(C) 2
(D) −2
Answer: (B) −½

• Order and Degree Important Questions

Order is based on the highest derivative present, while degree is determined from the highest-order derivative after the equation is made free from radicals and fractions with respect to derivatives. The chapter also includes questions that test the number of arbitrary constants in a solution.

1. The order and degree of the differential equation xy(d²y/dx²) + x(dy/dx)² − y(dy/dx) = 0 is
(A) 1, 2
(B) 2, 2
(C) 2, 1
(D) 1, 1
Answer: (C) 2, 1

2. The order and degree of the differential equation (d³y/dx³) + (d²y/dx²) + e^(dy/dx) = 0 is
(A) 3, 1
(B) 3, 2
(C) 1, 3
(D) 3, not defined
Answer: (D) 3, not defined

3. Sum of the squares of the order and degree of a differential equation x(y'')³ + √(y') = 0 is [KCET 2026]
(A) 3
(B) 20
(C) 8
(D) 16
Answer: (B) 20

4. The sum of the degree and order of the differential equation (d²y/dx²)³ = 0 is [KCET 2022]
(A) 4
(B) 6
(C) 5
(D) 7
Answer: (C) 5

5. The degree of the differential equation (d²y/dx²)³ + (dy/dx)² √(3 + (d²y/dx²)) = 0 [KCET 2023]
(A) 1
(B) 2
(C) 2
(D) 6
Answer: (D) 6

6. The order & degree of the D.E. √(1 + (dy/dx)) = (d²y/dx²) are
(A) 1 & 1
(B) 1 & 2
(C) 2 & 1
(D) 2 & 2
Answer: (A) 1 & 1

7. The order and degree of the differential equation [(dy/dx)² + (d²y/dx²)]^(5/4) = k(d³y/dx³) is
(A) 5, 2
(B) 2, 5
(C) 3, 4
(D) 4, 3
Answer: (C) 3, 4

8. The order of the differential equation is always
(A) Rational number
(B) Whole number
(C) Negative integer
(D) Positive integer
Answer: (D) Positive integer

• Formation of Differential Equations Questions

The chapter explains the formation of differential equations by differentiating the given relation and eliminating arbitrary constants. The number of independent arbitrary constants in the solution corresponds to the order of the differential equation.

1. Find the number of arbitrary constants in the particular solution of a differential equation of third order
(A) 3
(B) 2
(C) 1
(D) 0
Answer: (D) 0

2. The number of arbitrary constants in the general solution of a differential equation of fourth order is
(A) 4
(B) 3
(C) 2
(D) 1
Answer: (A) 4

3. The number of arbitrary constants in the solution of a differential equation of a degree 3 and order 2 is
(A) 3
(B) 2
(C) 5
(D) 1
Answer: (B) 2

4. The order of a differential equation whose general solution is y = A sin x + B cos x is
(A) 4
(B) 2
(C) 0
(D) 3
Answer: (B) 2

5. If d/dx[f(x)] = ax + b and f(0) = 0, then f(x) is equal to
(A) (ax²/2) + bx + c
(B) a + b
(C) b
(D) (ax²/2) + bx
Answer: (D) (ax²/2) + bx

6. The number of solutions of dy/dx = (y+1)/(x−1), when y(1) = 2, is [KCET 2021]
(A) three
(B) one
(C) infinite
(D) two
Answer: (B) one

7. The equation y = mx + c is general solution of
(A) x(dy/dx) = y
(B) y(dy/dx) = x
(C) d²y/dx² = 0
(D) d²x/dy² = 0
Answer: (C) d²y/dx² = 0

• Solving Differential Equations Important Questions

The material covers variable separable, homogeneous and linear differential equations. For homogeneous equations, the substitution y = vx is used to convert the equation into variable separable form. Linear differential equations are solved using an integrating factor.

1. Solution of differential equation x dy − y dx = 0 represents [KCET 2021]
(A) A rectangular hyperbola
(B) Parabola whose vertex is at origin
(C) Straight line passing through origin
(D) A circle whose centre is origin
Answer: (C) Straight line passing through origin

2. The general solution of the differential equation (y dx − x dy)/y = 0 is
(A) xy = C
(B) x = Cy²
(C) y = Cx
(D) y = Cx²
Answer: (C) y = Cx

3. The solution of e^(dy/dx) = x + 1, y(0) = 3 is [KCET 2024]
(A) y − 2 = x log x − x
(B) y − x − 3 = x log x
(C) y − x − 3 = (x+1) log(x+1)
(D) y + x − 3 = (x+1) log(x+1)
Answer: (D) y + x − 3 = (x+1) log(x+1)

4. A differential equation of the form dy/dx = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree
(A) 1
(B) 0
(C) 2
(D) 2, not defined
Answer: (B) 0

5. A homogeneous differential equation of the form dy/dx = h(x/y) can be solved by making the substitution
(A) y = vx
(B) v = yx
(C) x = vy
(D) x = v
Answer: (C) x = vy

6. The solution of (1 + e^(x/y)) dx + e^(x/y)(1 − x/y) dy = 0 is
(A) y e^(y/x) + x = c
(B) y e^(x/y) − x = c
(C) y e^(x/y) + y = c
(D) y e^(x/y) + x = c
Answer: (D) y e^(x/y) + x = c

7. Integrating factor of the differential equation (1 + x²) dy/dx + xy = 1 is [KCET 2026]
(A) 1 + x²
(B) ½ log(1 + x²)
(C) x/(1+x²)
(D) √(1 + x²)
Answer: (D) √(1 + x²)

8. General solution of the differential equation dy/dx + y tan x = sec x is [KCET 2025]
(A) y sec x = tan x + c
(B) y tan x = sec x + c
(C) cosec x = y tan x + c
(D) x sec x = tan y + c
Answer: (A) y sec x = tan x + c

9. If dy/dx + y/x = x², then 2y(2) − y(1) = [KCET 2022]
(A) 11/4
(B) 15/4
(C) 9/4
(D) 13/4
Answer: (B) 15/4

10. The Integrating Factor of the differential equation dy/dx = x + xy is e^(kx²). Then k is
(A) ½
(B) −½
(C) 2
(D) −2
Answer: (B) −½

• Differential Equations MCQs

The chapter contains 60 multiple-choice questions covering order and degree, solutions, variable separable equations, homogeneous equations and linear differential equations. Questions are marked as Easy, Average or Difficult, and several are identified with KCET years such as 2021, 2022, 2024, 2025 and 2026.

1. If √x · ∛y = (x+y)^n and x(dy/dx) − y = 0, then n = [KCET 2026]
(A) 1
(B) 6/5
(C) 5/6
(D) 4/9
Answer: (C) 5/6

2. The family of curves whose x and y intercepts of a tangent at any point are respectively double the x and y coordinates of that point is [KCET 2024]
(A) xy = C
(B) x² + y² = C
(C) x² − y² = C
(D) y/x = C
Answer: (A) xy = C

3. The solution of the differential equation dy/dx = (x+y)² is [KCET 2022]
(A) tan⁻¹(x+y) = x + C
(B) tan⁻¹(x+y) = 0
(C) cot⁻¹(x+y) = C
(D) cot⁻¹(x+y) = x + C
Answer: (A) tan⁻¹(x+y) = x + C

4. Which of the following is a homogeneous differential equation?
(A) (4x + 6y + 5) dy − (3y + 2x + 4) dx = 0
(B) (xy) dx − (x³ + y³) dy = 0
(C) (x³ + 2y²) dx + 2xy dy = 0
(D) y² dx + (x² − xy − y²) dy = 0
Answer: (D) y² dx + (x² − xy − y²) dy = 0

5. The Integrating Factor of the differential equation eˣ dy + (y eˣ + 2x) dx = 0 is e^(kx). Then k is
(A) 1
(B) −1
(C) 2
(D) −2
Answer: (A) 1

6. Statement 1: The integrating factor of the differential equation dx/dy + Px = Q(y) is e^(∫P dy). Statement 2: The integrating factor of the differential equation dy/dx + Py = Q(x) is e^(∫P dy).
(A) Statement 1 is false and Statement 2 is true
(B) Statement 1 is true and Statement 2 is false
(C) Statement 1 is true, Statement 2 is true and Statement 2 is a correct explanation for Statement 1
(D) Statement 1 is true, Statement 2 is true and Statement 2 is not a correct explanation for Statement 1
Answer: (C)

7. General solution of the differential equation dx/dy + x/y = y² is
(A) xy = y⁴/4 + c
(B) xy = y³/3 + c
(C) x = y³/3 + c/y
(D) y = x⁴/4 + c
Answer: (A) xy = y⁴/4 + c

Karnataka 2nd PUC Maths Differential Equations Important Questions PDF

Students can use the Differential Equations Important Questions PDF for chapter-wise MCQ practice. The PDF contains the 2026-27 Mathematics CET material for Differential Equations and includes 60 questions with an answer key.

How to Use Important Solutions for Karnataka 2nd PUC Differential Equations

Regular problem-solving ensures better speed and high accuracy in the final exam. Follow these steps to prepare effectively with the provided solutions:

• Master Basic Definitions – Learn the exact definitions of order, degree, and arbitrary constants before solving complex equations.

• Practice Separation of Variables – Solve standard questions by separating variables directly to find general and particular solutions.

• Focus on Integrating Factors – Identify standard linear differential equation forms and calculate the integrating factor accurately.

• Solve Past Exam Papers – Practice Karnataka 2nd PUC Previous Year Question Paper

• and check the Karnataka 2nd PUC Previous Year Question Paper.

• Check Exam Pattern – Review the Karnataka 2nd PUC Exam Pattern 2027 to master Differential Equations 5 Mark Questions thoroughly.

Karnataka 2nd PUC Maths Chapter 9 covers important areas of Differential Equations, including order and degree, solution methods, homogeneous equations and linear differential equations. Practising the chapter MCQs and checking the answer key can support systematic revision.

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