A FIRST COURSE IN DIFFERENTIAL EQUATIONS (ZILL) | COMPLETE EXAM
REVIEW 2026/2027 | 100% VERIFIED SOLUTIONS | FIRST-ORDER TO LAPLACE
| PASS GUARANTEE
Question 1. Which of the following is the general solution of the differential
equation dy/dx = 3y?
(A) y = Ce^(3x)
(B) y = 3Ce^x
(C) y = Ce^x + 3
(D) y = 3x + C
ANSWER : (A)
Explanation: Separating variables: dy/y = 3 dx. Integrating both sides gives
ln|y| = 3x + C₁, so y = Ce^(3x). This is a standard separable ODE result.
Question 2. The order and degree of the differential equation d²y/dx² +
(dy/dx)³ + y = 0 are, respectively:
(A) Order 3, degree 2
(B) Order 2, degree 1
(C) Order 1, degree 3
(D) Order 2, degree 3
ANSWER : (B)
Explanation: The highest derivative is d²y/dx², so the order is 2. The degree is
the power of the highest derivative once the equation is polynomial in
derivatives — d²y/dx² appears to the first power, so the degree is 1.
Question 3. A differential equation of the form M(x, y)dx + N(x, y)dy = 0 is
exact if and only if:
(A) ∂M/∂x = ∂N/∂y
, (B) ∂M/∂y = ∂N/∂x
(C) M = N
(D) ∂²M/∂x² = ∂²N/∂y²
ANSWER : (B)
Explanation: The exactness condition requires ∂M/∂y = ∂N/∂x. This ensures
the existence of a potential function F(x, y) such that dF = Mdx + Ndy.
Question 4. The integrating factor for the linear first-order equation dy/dx
+ P(x)y = Q(x) is:
(A) μ(x) = e^(∫Q(x)dx)
(B) μ(x) = e^(P(x))
(C) μ(x) = e^(∫P(x)dx)
(D) μ(x) = ∫P(x)dx
ANSWER : (C)
Explanation: Multiplying through by μ(x) = e^(∫P(x)dx) transforms the left
side into d/dx[μ(x)y], making the equation directly integrable.
Question 5. Which of the following is a Bernoulli differential equation?
(A) dy/dx + xy = x²
(B) dy/dx + y = y²
(C) d²y/dx² + y = 0
(D) dy/dx = x + y
ANSWER : (B)
Explanation: A Bernoulli equation has the form dy/dx + P(x)y = Q(x)yⁿ (n ≠ 0,
1). Option (B) fits with P = 1, Q = 1, n = 2. The substitution v = y^(1-n)
linearises it.
Question 6. The general solution of the second-order homogeneous
equation y'' - 5y' + 6y = 0 is:
(A) y = C₁e^(2x) + C₂e^(3x)
REVIEW 2026/2027 | 100% VERIFIED SOLUTIONS | FIRST-ORDER TO LAPLACE
| PASS GUARANTEE
Question 1. Which of the following is the general solution of the differential
equation dy/dx = 3y?
(A) y = Ce^(3x)
(B) y = 3Ce^x
(C) y = Ce^x + 3
(D) y = 3x + C
ANSWER : (A)
Explanation: Separating variables: dy/y = 3 dx. Integrating both sides gives
ln|y| = 3x + C₁, so y = Ce^(3x). This is a standard separable ODE result.
Question 2. The order and degree of the differential equation d²y/dx² +
(dy/dx)³ + y = 0 are, respectively:
(A) Order 3, degree 2
(B) Order 2, degree 1
(C) Order 1, degree 3
(D) Order 2, degree 3
ANSWER : (B)
Explanation: The highest derivative is d²y/dx², so the order is 2. The degree is
the power of the highest derivative once the equation is polynomial in
derivatives — d²y/dx² appears to the first power, so the degree is 1.
Question 3. A differential equation of the form M(x, y)dx + N(x, y)dy = 0 is
exact if and only if:
(A) ∂M/∂x = ∂N/∂y
, (B) ∂M/∂y = ∂N/∂x
(C) M = N
(D) ∂²M/∂x² = ∂²N/∂y²
ANSWER : (B)
Explanation: The exactness condition requires ∂M/∂y = ∂N/∂x. This ensures
the existence of a potential function F(x, y) such that dF = Mdx + Ndy.
Question 4. The integrating factor for the linear first-order equation dy/dx
+ P(x)y = Q(x) is:
(A) μ(x) = e^(∫Q(x)dx)
(B) μ(x) = e^(P(x))
(C) μ(x) = e^(∫P(x)dx)
(D) μ(x) = ∫P(x)dx
ANSWER : (C)
Explanation: Multiplying through by μ(x) = e^(∫P(x)dx) transforms the left
side into d/dx[μ(x)y], making the equation directly integrable.
Question 5. Which of the following is a Bernoulli differential equation?
(A) dy/dx + xy = x²
(B) dy/dx + y = y²
(C) d²y/dx² + y = 0
(D) dy/dx = x + y
ANSWER : (B)
Explanation: A Bernoulli equation has the form dy/dx + P(x)y = Q(x)yⁿ (n ≠ 0,
1). Option (B) fits with P = 1, Q = 1, n = 2. The substitution v = y^(1-n)
linearises it.
Question 6. The general solution of the second-order homogeneous
equation y'' - 5y' + 6y = 0 is:
(A) y = C₁e^(2x) + C₂e^(3x)
