Linear differential equations MCQs With Answer

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Linear differential equations MCQs With Answer for B. Pharm students offers a focused, applied review of linear ODE concepts used in pharmaceutical mathematics and pharmacokinetics. This introduction covers first- and higher-order linear equations, integrating factors, constant-coefficient methods, superposition, Wronskian, and solution techniques like undetermined coefficients and variation of parameters. Emphasis is on practical drug modeling: one- and multi-compartment systems, first-order elimination, half-life, steady state, and dose-response modeling. Clear examples and targeted MCQs build problem-solving skills essential for B. Pharm coursework and exams. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. Which of the following is the standard form of a first-order linear differential equation?

  • dy/dx + P(x) y = Q(x)
  • dy/dx = P(y)
  • y” + p(x) y’ + q(x) y = 0
  • y’ = ay + by^2

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

Q2. What is the integrating factor μ(x) for dy/dx + P(x)y = Q(x)?

  • 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}

Q3. The general solution of the homogeneous first-order equation dy/dx + P(x)y = 0 is:

  • y = C e^{∫P(x) dx}
  • y = C e^{-∫P(x) dx}
  • y = C + ∫P(x) dx
  • y = Ce^{∫Q(x) dx}

Correct Answer: y = C e^{-∫P(x) dx}

Q4. For a linear second-order ODE with constant coefficients ay” + by’ + cy = 0, the characteristic equation is:

  • aλ^2 + bλ + c = 0
  • a + bλ + cλ^2 = 0
  • λ + aλ^2 + b = 0
  • λ^2 + aλ + b = 0

Correct Answer: aλ^2 + bλ + c = 0

Q5. The superposition principle applies to which type of differential equations?

  • Nonlinear inhomogeneous equations
  • Homogeneous linear differential equations
  • Any first-order equation
  • Equations with variable coefficients only

Correct Answer: Homogeneous linear differential equations

Q6. Method of undetermined coefficients is best suited when the nonhomogeneous term is:

  • An arbitrary continuous function
  • Polynomial, exponential, sinusoidal, or product of these
  • A function with essential singularities
  • A random discontinuous input

Correct Answer: Polynomial, exponential, sinusoidal, or product of these

Q7. Variation of parameters is primarily used to:

  • Find complementary function for constant coefficients
  • Find particular solution when undetermined coefficients fail
  • Compute eigenvalues of a system
  • Transform a nonlinear ODE to linear form

Correct Answer: Find particular solution when undetermined coefficients fail

Q8. The Wronskian of two solutions y1 and y2 being nonzero at some point implies:

  • y1 and y2 are linearly dependent
  • y1 and y2 are linearly independent
  • The ODE is nonlinear
  • The ODE has no solution

Correct Answer: y1 and y2 are linearly independent

Q9. The order of a differential equation is defined by:

  • The highest power of the dependent variable
  • The highest derivative present in the equation
  • The degree of the polynomial coefficients
  • The number of independent variables

Correct Answer: The highest derivative present in the equation

Q10. The degree of a differential equation is:

  • The exponent of the highest derivative when the equation is a polynomial in derivatives
  • The order plus one
  • Always equal to the order for linear equations
  • The number of independent solutions

Correct Answer: The exponent of the highest derivative when the equation is a polynomial in derivatives

Q11. Existence and uniqueness for first-order linear ODE dy/dx + P(x)y = Q(x) require P and Q to be:

  • Discontinuous at x0
  • Continuous on an interval containing x0
  • Constant functions only
  • Polynomials of degree ≤ 2

Correct Answer: Continuous on an interval containing x0

Q12. For a repeated root r of multiplicity 2 of the characteristic equation, the general solution is:

  • y = C1 e^{rx} + C2 e^{2rx}
  • y = C1 e^{rx} + C2 x e^{rx}
  • y = C1 cos(rx) + C2 sin(rx)
  • y = C1 e^{rx} only

Correct Answer: y = C1 e^{rx} + C2 x e^{rx}

Q13. If the characteristic roots are a ± bi, the real general solution is:

  • y = e^{ax}[C1 cos(bx) + C2 sin(bx)]
  • y = C1 e^{(a+bi)x} + C2 e^{(a-bi)x}
  • y = C1 cosh(ax) + C2 sinh(bx)
  • y = e^{bx}[C1 cos(ax) + C2 sin(ax)]

Correct Answer: y = e^{ax}[C1 cos(bx) + C2 sin(bx)]

Q14. Laplace transforms are particularly useful for solving linear ODEs when:

  • Initial conditions are zero only
  • Dealing with piecewise or impulse (delta) inputs and initial conditions
  • The equation is nonlinear
  • Coefficients are not integrable

Correct Answer: Dealing with piecewise or impulse (delta) inputs and initial conditions

Q15. A differential equation is linear if:

  • The dependent variable and its derivatives appear only to the first power and are not multiplied together
  • It contains exponential terms in the coefficients
  • The coefficients are constants only
  • Derivatives appear at integer orders only

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

Q16. In pharmacokinetics, a one-compartment IV bolus model follows which linear ODE for concentration C(t)?

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

Correct Answer: dC/dt = -k C

Q17. For first-order elimination, the half-life t1/2 is given by:

  • t1/2 = k / ln 2
  • t1/2 = ln 2 / k
  • t1/2 = 2 / k
  • t1/2 = ln(k)/2

Correct Answer: t1/2 = ln 2 / k

Q18. The concentration solution for first-order elimination with initial concentration C0 is:

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

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

Q19. For continuous infusion at rate R_in and clearance Cl, steady-state concentration Css is:

  • Css = R_in × Cl
  • Css = R_in / Cl
  • Css = Cl / R_in
  • Css = R_in × t1/2

Correct Answer: Css = R_in / Cl

Q20. Reduction of order is a method used to:

  • Solve nonlinear algebraic equations
  • Find a second independent solution of a second-order linear ODE given one solution
  • Compute integrating factors for first-order ODEs
  • Determine stability of fixed points only

Correct Answer: Find a second independent solution of a second-order linear ODE given one solution

Q21. If a particular solution trial duplicates a homogeneous solution, the correct modification is to:

  • Divide the trial by x
  • Multiply the trial by an appropriate power of x
  • Use Laplace transforms instead
  • Discard the trial and use numerical methods

Correct Answer: Multiply the trial by an appropriate power of x

Q22. Applying the integrating factor transforms dy/dx + P(x)y = Q(x) into:

  • d/dx[μ(x) y] = μ(x) Q(x)
  • μ(x) dy/dx = Q(x)
  • d/dx[y/μ(x)] = Q(x)
  • μ(x) d^2y/dx^2 = Q(x)

Correct Answer: d/dx[μ(x) y] = μ(x) Q(x)

Q23. The solution of a linear system x’ = A x uses which matrix function?

  • Determinant det(A t)
  • Matrix exponential e^{A t}
  • Inverse matrix A^{-t}
  • Trace of A

Correct Answer: Matrix exponential e^{A t}

Q24. Linearity implies which two properties for solutions y1 and y2 of the homogeneous equation?

  • Only addition holds
  • Superposition: a y1 + b y2 is also a solution for scalars a, b
  • Solutions cannot be scaled
  • Only multiplication by functions is allowed

Correct Answer: Superposition: a y1 + b y2 is also a solution for scalars a, b

Q25. Undetermined coefficients can directly solve y” + y = sin x by assuming a particular solution of the form:

  • y_p = A e^{x}
  • y_p = A x + B
  • y_p = A sin x + B cos x
  • y_p = A x e^{x}

Correct Answer: y_p = A sin x + B cos x

Q26. The general solution of an nth-order linear ODE equals:

  • Only the particular integral
  • Sum of complementary function (homogeneous solution) and particular integral
  • Product of solutions of first-order equations
  • Only the homogeneous solution

Correct Answer: Sum of complementary function (homogeneous solution) and particular integral

Q27. Multi-compartment pharmacokinetic models with first-order transfers yield:

  • Nonlinear algebraic equations
  • Systems of linear ODEs
  • Partial differential equations only
  • Difference equations only

Correct Answer: Systems of linear ODEs

Q28. Stability of an equilibrium in a linear system x’ = A x is determined by:

  • Determinant of A alone
  • Eigenvalues of A (their real parts)
  • Trace of A only
  • Rank of A only

Correct Answer: Eigenvalues of A (their real parts)

Q29. For dy/dt + p(t) y = q(t) with p and q continuous everywhere, the initial value problem has:

  • No solution
  • At most one discontinuous solution
  • A unique solution on the interval
  • Infinitely many solutions passing through a point

Correct Answer: A unique solution on the interval

Q30. Green’s function is used to represent solutions of linear inhomogeneous ODEs subject to:

  • Algebraic constraints only
  • Boundary value problems or linear operators with specified boundary conditions
  • Only homogeneous initial conditions
  • Nonlinear boundary conditions

Correct Answer: Boundary value problems or linear operators with specified boundary conditions

Q31. If the Wronskian of two solutions is zero for all x in an interval, then:

  • The two solutions are linearly independent
  • The two solutions are linearly dependent
  • The ODE is non-linear
  • The interval contains a singularity

Correct Answer: The two solutions are linearly dependent

Q32. What is the degree of a linear differential equation (when degree is defined)?

  • Equal to the order
  • One
  • Zero
  • Depends on coefficients

Correct Answer: One

Q33. For RHS of form x^n e^{ax}, the undetermined coefficients trial should be:

  • A polynomial of degree n times e^{ax}
  • An exponential only
  • Sine-cosine combination
  • A rational function

Correct Answer: A polynomial of degree n times e^{ax}

Q34. The integrating factor exists for dy/dx + p(x) y = q(x) provided:

  • p(x) is integrable on the interval
  • q(x) is zero
  • p(x) is constant only
  • The equation is homogeneous

Correct Answer: p(x) is integrable on the interval

Q35. The Frobenius method is used to find series solutions near a point when the ODE has:

  • A regular singular point
  • No singularities
  • Essential singularities only
  • Constant coefficients only

Correct Answer: A regular singular point

Q36. The operator L[y] = y” + a1 y’ + a0 y is linear because:

  • L[ay1 + by2] = a L[y1] + b L[y2]
  • L[y1 y2] = L[y1] L[y2]
  • L[y] multiplies y by constant only
  • L[y] depends nonlinearly on derivatives

Correct Answer: L[ay1 + by2] = a L[y1] + b L[y2]

Q37. Nontrivial solutions of homogeneous linear boundary value problems typically occur at:

  • Arbitrary parameter values
  • Discrete eigenvalues
  • Only for zero boundary conditions
  • Only in first-order problems

Correct Answer: Discrete eigenvalues

Q38. For linear ODEs with constant coefficients, the form of solutions generally includes:

  • Polynomials only
  • Exponential functions, and possibly sines and cosines
  • Logarithmic functions only
  • Rational functions only

Correct Answer: Exponential functions, and possibly sines and cosines

Q39. Variation of parameters formula to find particular solution uses which determinant in its denominator?

  • Jacobian determinant of independent variables
  • The Wronskian of the fundamental solutions
  • Determinant of coefficient matrix only for constant coefficients
  • Determinant of boundary conditions

Correct Answer: The Wronskian of the fundamental solutions

Q40. Laplace transforms convert linear ODEs into:

  • Algebraic equations in the Laplace domain
  • Higher-order differential equations
  • Nonlinear integral equations only
  • Partial differential equations

Correct Answer: Algebraic equations in the Laplace domain

Q41. The solution space dimension for an nth-order linear homogeneous ODE is:

  • 1 for all n
  • n
  • Depends on coefficients
  • Infinity

Correct Answer: n

Q42. Reduction of order requires which initial knowledge?

  • A particular solution for the nonhomogeneous equation
  • A fundamental matrix for systems
  • One nontrivial solution of the homogeneous equation
  • Boundary conditions at two points

Correct Answer: One nontrivial solution of the homogeneous equation

Q43. An equation containing y^2 or y y’ is classified as:

  • Linear
  • Nonlinear
  • Constant coefficient linear
  • Separable linear

Correct Answer: Nonlinear

Q44. Constant-coefficient linear ODE solutions often take the form:

  • Power series with variable exponents only
  • Exponentials e^{rx}, or sines/cosines when roots are complex
  • Rational functions of x only
  • Logarithmic-exponential hybrids only

Correct Answer: Exponentials e^{rx}, or sines/cosines when roots are complex

Q45. For a linear system x’ = A x, if all eigenvalues of A have negative real parts, the equilibrium at 0 is:

  • Unstable
  • Asymptotically stable
  • Marginally stable
  • Periodic

Correct Answer: Asymptotically stable

Q46. Modeling an IV bolus as a Dirac delta input in a linear ODE is often handled by:

  • Numerical finite differences only
  • Laplace transforms to incorporate the impulse directly
  • Discarding initial conditions
  • Assuming steady-state instantly

Correct Answer: Laplace transforms to incorporate the impulse directly

Q47. The general solution from integrating factor method can be written as:

  • y = e^{∫P} (∫e^{-∫P} Q dx + C)
  • y = e^{-∫P} (∫e^{∫P} Q dx + C)
  • y = ∫P dx + ∫Q dx
  • y = C e^{∫Q dx}

Correct Answer: y = e^{-∫P} (∫e^{∫P} Q dx + C)

Q48. To satisfy an initial condition y(x0)=y0 for a general solution with constant C, one must:

  • Differentiate and solve for C
  • Substitute x0 and y0 into the general solution and solve for C
  • Set C = 0 always
  • Use boundary conditions instead

Correct Answer: Substitute x0 and y0 into the general solution and solve for C

Q49. Given y” + 4y’ + 4y = x e^{-2x}, because e^{-2x} is a solution of the homogeneous equation with multiplicity 2, the appropriate form of particular solution is:

  • y_p = (Ax + B) e^{-2x}
  • y_p = (Ax^2 + Bx + C) e^{-2x}
  • y_p = A e^{-2x}
  • y_p = (Ax^3 + Bx^2) e^{-2x}

Correct Answer: y_p = (Ax^2 + Bx + C) e^{-2x}

Q50. A key implication of linear (first-order) elimination kinetics in pharmacology is that half-life is:

  • Dependent on dose
  • Independent of dose
  • Proportional to dose squared
  • Undefined for constant clearance

Correct Answer: Independent of dose

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