Differential Equations – Definitions MCQs With Answer

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Differential Equations – Definitions MCQs With Answer provide B. Pharm students a focused way to master core concepts used in pharmacokinetics and drug modeling. This set covers fundamental definitions—order, degree, ordinary vs. partial, linear vs. nonlinear—along with methods like separation of variables, integrating factors, exact equations, Bernoulli forms and Laplace transforms. Emphasis is on practical applications in drug elimination, compartmental models, half-life, clearance and steady-state. These concise, exam-oriented MCQs strengthen problem-solving skills and conceptual clarity required for coursework and practical pharmacy problems. Use these questions to review terminology, solution methods and real-world pharmacological models. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is a differential equation?

  • An equation relating functions and their derivatives
  • An algebraic equation with only constants
  • A statistical model for drug trials
  • A chemical reaction rate law

Correct Answer: An equation relating functions and their derivatives

Q2. In a differential equation, what does the “order” refer to?

  • The highest power of the dependent variable
  • The highest derivative present in the equation
  • The number of independent variables
  • The degree after clearing fractions

Correct Answer: The highest derivative present in the equation

Q3. How is the “degree” of a differential equation defined?

  • The highest power of the highest order derivative after clearing fractions and radicals
  • The number of independent variables
  • The number of arbitrary constants in the general solution
  • The coefficient of the highest derivative

Correct Answer: The highest power of the highest order derivative after clearing fractions and radicals

Q4. What distinguishes an ordinary differential equation (ODE) from a partial differential equation (PDE)?

  • An ODE involves derivatives with respect to a single independent variable; PDEs involve partial derivatives with respect to multiple independent variables
  • An ODE is always linear while a PDE is nonlinear
  • An ODE cannot model pharmacokinetics
  • PDEs have no real-world applications

Correct Answer: An ODE involves derivatives with respect to a single independent variable; PDEs involve partial derivatives with respect to multiple independent variables

Q5. What defines a linear differential equation?

  • The dependent variable and its derivatives appear only to the first power and are not multiplied together
  • The equation has constant coefficients only
  • The equation is separable into two functions
  • The solution is always exponential

Correct Answer: The dependent variable and its derivatives appear only to the first power and are not multiplied together

Q6. A linear differential equation is homogeneous when:

  • The equation contains no derivatives
  • The right-hand side is zero (no forcing term)
  • The coefficients are variable
  • It has repeated roots in the characteristic equation

Correct Answer: The right-hand side is zero (no forcing term)

Q7. What is the general solution of a differential equation?

  • A specific numerical solution at a point
  • The family of solutions that includes arbitrary constants equal to the order of the equation
  • A solution with no arbitrary constants
  • A constant function only

Correct Answer: The family of solutions that includes arbitrary constants equal to the order of the equation

Q8. What is a particular solution?

  • A solution of the homogeneous equation only
  • A specific member of the general solution that satisfies given initial or boundary conditions
  • The general solution minus arbitrary constants
  • A solution that does not satisfy any initial condition

Correct Answer: A specific member of the general solution that satisfies given initial or boundary conditions

Q9. What describes an initial value problem (IVP)?

  • An ODE with boundary conditions specified at two points
  • An ODE together with values of the unknown function (and possibly derivatives) at a single point
  • A PDE with no conditions
  • An algebraic equation paired with initial guesses

Correct Answer: An ODE together with values of the unknown function (and possibly derivatives) at a single point

Q10. What is a boundary value problem (BVP)?

  • An ODE with conditions specified at one point only
  • An ODE or PDE with conditions prescribed at two or more points of the domain
  • A statistical boundary for experimental error
  • Any nonlinear differential equation

Correct Answer: An ODE or PDE with conditions prescribed at two or more points of the domain

Q11. Which equation is separable?

  • dy/dx = x + y
  • dy/dx = x*y
  • dy/dx = f(x)g(y) so variables can be separated into functions of x and y
  • dy/dx + P(x)y = Q(x)

Correct Answer: dy/dx = f(x)g(y) so variables can be separated into functions of x and y

Q12. What is the integrating factor method used for?

  • Solving exact equations only
  • Solving first-order linear ODEs of the form dy/dx + P(x)y = Q(x)
  • Finding eigenvalues of matrices
  • Transforming PDEs to ODEs

Correct Answer: Solving first-order linear ODEs of the form dy/dx + P(x)y = Q(x)

Q13. When is M(x,y)dx + N(x,y)dy = 0 exact?

  • When ∂M/∂x = ∂N/∂y
  • When ∂M/∂y = ∂N/∂x
  • When M and N are constants
  • When M(x,y) = N(x,y)

Correct Answer: When ∂M/∂y = ∂N/∂x

Q14. The Bernoulli differential equation has which general form?

  • dy/dx + P(x)y = Q(x)y^n
  • d^2y/dx^2 + ay + b = 0
  • dy/dx = f(x) + g(y)
  • y” + p(x)y’ + q(x)y = r(x)

Correct Answer: dy/dx + P(x)y = Q(x)y^n

Q15. For a second-order linear ODE with constant coefficients, the characteristic equation is formed by replacing derivatives with:

  • sinusoidal functions
  • powers of t
  • r (so y” → r^2, y’ → r, y → 1)
  • logarithms

Correct Answer: r (so y” → r^2, y’ → r, y → 1)

Q16. What is the complementary function (CF) of a linear ODE?

  • The particular solution for the nonhomogeneous equation
  • The general solution of the associated homogeneous equation
  • The constant coefficient of the differential operator
  • The Laplace transform of the solution

Correct Answer: The general solution of the associated homogeneous equation

Q17. What is the particular integral (PI) in solving nonhomogeneous linear ODEs?

  • A solution of the homogeneous equation only
  • A specific solution of the nonhomogeneous equation without arbitrary constants
  • The complementary function multiplied by constants
  • The Fourier transform of the solution

Correct Answer: A specific solution of the nonhomogeneous equation without arbitrary constants

Q18. What does a nonzero Wronskian of two solutions indicate?

  • The solutions are linearly dependent
  • The solutions are linearly independent
  • The equation has no solution
  • The solutions are identical

Correct Answer: The solutions are linearly independent

Q19. The superposition principle applies to which type of equations?

  • Linear homogeneous differential equations
  • Nonlinear differential equations only
  • Any separable equation
  • Only second-order equations

Correct Answer: Linear homogeneous differential equations

Q20. For an autonomous ODE dy/dt = f(y), an equilibrium point y* satisfies:

  • f'(y*) = 0
  • f(y*) = 0
  • y* = 0 only
  • f(y*) = y*

Correct Answer: f(y*) = 0

Q21. In pharmacokinetics, first-order elimination means:

  • Elimination rate is constant over time
  • Elimination rate is proportional to the drug concentration
  • Drug is eliminated only after a threshold concentration
  • Elimination follows zero-order kinetics

Correct Answer: Elimination rate is proportional to the drug concentration

Q22. Zero-order kinetics describes drug elimination where:

  • Rate is proportional to concentration
  • Rate is constant independent of concentration
  • Drug accumulates exponentially
  • Rate depends on square of concentration

Correct Answer: Rate is constant independent of concentration

Q23. For first-order elimination, the half-life t1/2 equals:

  • ln(2) × k
  • ln(2) / k
  • k / ln(2)
  • 2k

Correct Answer: ln(2) / k

Q24. Clearance (CL) relates elimination rate constant k and volume of distribution Vd by:

  • CL = Vd / k
  • CL = k / Vd
  • CL = k × Vd
  • CL = Vd − k

Correct Answer: CL = k × Vd

Q25. Volume of distribution Vd is defined (instantaneous distribution) as:

  • Vd = Dose / C0
  • Vd = C0 / Dose
  • Vd = Dose × CL
  • Vd = CL / k

Correct Answer: Vd = Dose / C0

Q26. The one-compartment model with first-order elimination is described by which differential equation for concentration C?

  • dC/dt = kC
  • dC/dt = -kC
  • dC/dt = -kC^2
  • dC/dt = k

Correct Answer: dC/dt = -kC

Q27. The solution of dC/dt = -kC with initial concentration C0 is:

  • C = C0 + kt
  • C = C0 e^{kt}
  • C = C0 e^{-kt}
  • C = k / (1 + kt)

Correct Answer: C = C0 e^{-kt}

Q28. Steady-state during continuous dosing is typically approached after approximately how many half-lives?

  • 1–2 half-lives
  • 4–5 half-lives
  • 10–12 half-lives
  • Less than one half-life

Correct Answer: 4–5 half-lives

Q29. Why are Laplace transforms useful for initial value problems?

  • They convert algebraic equations to differential equations
  • They convert differential equations to algebraic equations incorporating initial conditions
  • They eliminate the need for constants
  • They only work for nonlinear ODEs

Correct Answer: They convert differential equations to algebraic equations incorporating initial conditions

Q30. The explicit Euler method for numerical solution of dy/dt = f(t,y) has local truncation error of which order?

  • Zero order
  • First order
  • Second order
  • Fourth order

Correct Answer: First order

Q31. The classical Runge–Kutta method of order 4 (RK4) has global error of what order?

  • First order
  • Second order
  • Third order
  • Fourth order

Correct Answer: Fourth order

Q32. The logistic growth equation dN/dt = rN(1 − N/K) is an example of which type of differential equation?

  • Linear homogeneous ODE
  • Nonlinear ODE
  • Second-order linear ODE
  • Exact differential equation

Correct Answer: Nonlinear ODE

Q33. An equilibrium (steady-state) solution y* of dy/dt = f(y) is:

  • A solution that increases without bound
  • A constant solution satisfying f(y*) = 0
  • A solution dependent on t explicitly
  • Only for linear systems

Correct Answer: A constant solution satisfying f(y*) = 0

Q34. If the characteristic equation has a repeated root r of multiplicity 2, the general solution contains which term?

  • e^{rt} and independently sin(rt)
  • e^{rt} and t e^{rt}
  • Only e^{rt} twice
  • Only polynomial solutions

Correct Answer: e^{rt} and t e^{rt}

Q35. For a sinusoidal forcing term in a linear ODE, the method of undetermined coefficients suggests a particular solution of what form?

  • An exponential only
  • A polynomial only
  • A sinusoid (sine and cosine) of the same frequency
  • A logarithmic function

Correct Answer: A sinusoid (sine and cosine) of the same frequency

Q36. Which property defines a linear differential operator L acting on functions?

  • L(af + bg) = aL(f) + bL(g) for scalars a, b and functions f, g
  • L(fg) = L(f)L(g)
  • L(f+g) = L(f) × L(g)
  • L always has constant coefficients

Correct Answer: L(af + bg) = aL(f) + bL(g) for scalars a, b and functions f, g

Q37. An integrating factor μ(x) for dy/dx + P(x)y = Q(x) is given by:

  • μ(x) = e^{∫Q(x)dx}
  • μ(x) = e^{∫P(x)dx}
  • μ(x) = ∫P(x)dx
  • μ(x) = P(x)Q(x)

Correct Answer: μ(x) = e^{∫P(x)dx}

Q38. Existence and uniqueness theorem for dy/dx = f(x,y) requires which condition on f with respect to y?

  • f must be discontinuous in y
  • f must satisfy a Lipschitz condition in y (or continuous ∂f/∂y)
  • f must be unbounded
  • f must be independent of x

Correct Answer: f must satisfy a Lipschitz condition in y (or continuous ∂f/∂y)

Q39. A second-order ODE missing the independent variable x (i.e., y” = f(y,y’)) can often be reduced by the substitution:

  • y = vx where v depends on x
  • Let p = y’ and consider p as a function of y, reducing order
  • Use Laplace transforms directly
  • Differentiate with respect to x again

Correct Answer: Let p = y’ and consider p as a function of y, reducing order

Q40. A homogeneous differential equation of the form dy/dx = F(y/x) suggests substitution:

  • y = vx (so v = y/x)
  • x = vy
  • y = e^{vx}
  • Use integrating factor directly

Correct Answer: y = vx (so v = y/x)

Q41. Method of undetermined coefficients is suitable when the nonhomogeneous term is:

  • Any arbitrary function
  • A polynomial, exponential, sine or cosine, or their combinations
  • A rapidly varying stochastic function
  • A function involving integrals of the solution

Correct Answer: A polynomial, exponential, sine or cosine, or their combinations

Q42. For a linear system x’ = Ax, stability of the zero solution is determined by:

  • Determinant of A only
  • Eigenvalues of A; negative real parts imply asymptotic stability
  • Trace of A only
  • Whether A is symmetric

Correct Answer: Eigenvalues of A; negative real parts imply asymptotic stability

Q43. In a one-dimensional autonomous system, arrows on the phase line point toward an equilibrium if:

  • The derivative f(y) > 0
  • The derivative f(y) < 0
  • The derivative f(y) = 0 everywhere
  • The independent variable is time

Correct Answer: The derivative f(y) < 0

Q44. Green’s functions are primarily used to solve which type of problems?

  • Nonlinear initial value problems
  • Linear boundary value problems
  • Algebraic equations only
  • Stochastic differential equations

Correct Answer: Linear boundary value problems

Q45. A singular solution of a differential equation is:

  • A member of the general solution family for all parameter values
  • An envelope of the family of general solutions not obtainable by choosing constants
  • Always the trivial zero solution
  • Only relevant for linear equations

Correct Answer: An envelope of the family of general solutions not obtainable by choosing constants

Q46. Picard iteration method is used to:

  • Find exact algebraic solutions only
  • Construct successive approximations to the solution of an IVP
  • Solve boundary value problems directly
  • Compute eigenvalues numerically

Correct Answer: Construct successive approximations to the solution of an IVP

Q47. Separation of variables is a standard method for solving which PDE-related technique?

  • Transforming integral equations to algebraic ones
  • Reducing PDEs to ODEs by assuming product solutions in independent variables
  • Finding Green’s functions only
  • Calculating stochastic integrals

Correct Answer: Reducing PDEs to ODEs by assuming product solutions in independent variables

Q48. For a constant-rate IV infusion R0 and clearance CL, steady-state concentration Css equals:

  • Css = R0 × CL
  • Css = R0 / CL
  • Css = CL / R0
  • Css = R0 × Vd

Correct Answer: Css = R0 / CL

Q49. For the amount A in the body with input rate R and first-order elimination k, the steady-state amount A_ss satisfies:

  • A_ss = 0
  • A_ss = R / k
  • A_ss = k / R
  • A_ss = R × k

Correct Answer: A_ss = R / k

Q50. For immediate oral absorption with bioavailability F, the initial concentration C0 (instantaneous distribution) equals:

  • C0 = Dose / (F × Vd)
  • C0 = F × Dose × Vd
  • C0 = (F × Dose) / Vd
  • C0 = Vd / (F × Dose)

Correct Answer: C0 = (F × Dose) / Vd

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