Exact differential equations MCQs With Answer

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Exact differential equations MCQs With Answer are essential for B.Pharm students studying mathematical methods applied to pharmacokinetics and drug formulation. This concise introduction covers key concepts like the exactness condition (∂M/∂y = ∂N/∂x), potential (or scalar) functions, and integrating factors (μ(x) or μ(y)). These multiple-choice questions emphasize problem-solving skills, recognizing when an equation is exact, finding integrating factors, and interpreting solutions in pharmacy contexts such as compartmental models. Designed for exam preparation and practical application, these MCQs reinforce analytical reasoning and mathematical fluency needed in pharmaceutical sciences. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the condition for the first-order differential equation M(x,y)dx + N(x,y)dy = 0 to be exact?

  • The equation is linear
  • ∂M/∂y = ∂N/∂x
  • M and N are both functions of x only
  • ∂M/∂x = ∂N/∂y

Correct Answer: ∂M/∂y = ∂N/∂x

Q2. If M(x,y) = 2xy and N(x,y) = x^2 + 3y^2, is Mdx + Ndy = 0 exact?

  • Yes, because ∂M/∂y = ∂N/∂x
  • No, because ∂M/∂y ≠ ∂N/∂x
  • Yes, because M and N are homogeneous
  • No, because degree of M and N differ

Correct Answer: No, because ∂M/∂y ≠ ∂N/∂x

Q3. For an exact equation Mdx + Ndy = 0 with potential function φ(x,y), which relations hold?

  • φ_x = N and φ_y = M
  • φ_x = M and φ_y = N
  • φ = M + N
  • φ_x = φ_y

Correct Answer: φ_x = M and φ_y = N

Q4. To solve an exact equation, the general solution is given by:

  • Integrate M with respect to y and set equal to constant
  • Find φ such that φ(x,y) = C where φ_x = M and φ_y = N
  • Find integrating factor and integrate both sides separately
  • Use separation of variables

Correct Answer: Find φ such that φ(x,y) = C where φ_x = M and φ_y = N

Q5. If M(x,y) = y cos x and N(x,y) = sin x + x, does Mdx + Ndy = 0 satisfy exactness?

  • Yes, because ∂M/∂y = cos x and ∂N/∂x = cos x + 1
  • No, because ∂M/∂y ≠ ∂N/∂x
  • Yes, because both are functions of x only
  • No, because N is not differentiable

Correct Answer: No, because ∂M/∂y ≠ ∂N/∂x

Q6. Which integrating factor μ(x) depending only on x can make a non-exact Mdx + Ndy = 0 exact?

  • μ(x) = exp(∫[(∂N/∂x – ∂M/∂y)/M] dy)
  • μ(x) = exp(∫[(∂M/∂y – ∂N/∂x)/N] dx)
  • μ(x) = exp(∫M dx)
  • μ(x) = N(x,y)

Correct Answer: μ(x) = exp(∫[(∂M/∂y – ∂N/∂x)/N] dx)

Q7. Which integrating factor μ(y) depending only on y is used to make a differential equation exact?

  • μ(y) = exp(∫[(∂M/∂y – ∂N/∂x)/N] dx)
  • μ(y) = exp(∫[(∂N/∂x – ∂M/∂y)/M] dy)
  • μ(y) = exp(∫N dy)
  • μ(y) = M(x,y)

Correct Answer: μ(y) = exp(∫[(∂N/∂x – ∂M/∂y)/M] dy)

Q8. For M = (2x + y) and N = (x + 3y^2), ∂M/∂y = 1 and ∂N/∂x = 1. What is the type of this equation?

  • Non-exact
  • Exact
  • Homogeneous but not exact
  • Linear in y

Correct Answer: Exact

Q9. When using potential function method, after integrating M with respect to x you must:

  • Add a function of x only
  • Add a function of y only
  • Differentiate with respect to x again
  • Set the result equal to zero

Correct Answer: Add a function of y only

Q10. For M(x,y) = 3x^2y and N(x,y) = x^3 + 2y, check exactness: ∂M/∂y = 3x^2, ∂N/∂x = 3x^2. What is the general solution?

  • φ(x,y) = x^3 y + y^2 + C
  • φ(x,y) = x^3 + y^2 + C
  • φ(x,y) = x^3 y + 2y + C
  • φ(x,y) = 3x^2 y + C

Correct Answer: φ(x,y) = x^3 y + y^2 + C

Q11. In pharmacokinetics, an exact differential equation can appear when modeling:

  • Linear algebraic equations for drug solubility
  • Two-compartment models with conservation relations
  • Simple dilution without rates
  • Static equilibrium of tablets

Correct Answer: Two-compartment models with conservation relations

Q12. If an equation becomes exact after multiplying by x^n y^m (a power law integrating factor), this factor depends on:

  • Both x and y
  • Only x
  • Only y
  • Neither x nor y

Correct Answer: Both x and y

Q13. When is the integrating factor μ(x) = x^m y^n particularly useful?

  • When M and N are linear functions
  • When M and N are homogeneous functions of same degree
  • When equation is already exact
  • When equation contains exponential terms only

Correct Answer: When M and N are homogeneous functions of same degree

Q14. For M(x,y) = y^2 and N(x,y) = 2xy, check exactness: ∂M/∂y = 2y, ∂N/∂x = 2y. The general solution is:

  • φ = y^3/3 + x y^2 + C
  • φ = x y^2 + C
  • φ = y^3/3 + C
  • φ = x^2 y + C

Correct Answer: φ = x y^2 + y^3/3 + C

Q15. For an equation Mdx + Ndy = 0 that is not exact, which procedure is correct?

  • Claim no solution exists
  • Find an integrating factor to make it exact
  • Differentiate M and N and equate them
  • Convert it to a PDE

Correct Answer: Find an integrating factor to make it exact

Q16. Given M = 2xy + y^2 and N = x^2 + 2xy, ∂M/∂y = 2x + 2y, ∂N/∂x = 2x + 2y. The potential φ(x,y) is:

  • φ = x^2 y + x y^2 + C
  • φ = x y + C
  • φ = x^2 + y^2 + C
  • φ = 2xy + C

Correct Answer: φ = x^2 y + x y^2 + C

Q17. If (∂M/∂y – ∂N/∂x)/N is a function of x alone, then:

  • An integrating factor μ(y) exists
  • An integrating factor μ(x) exists
  • No integrating factor exists
  • The equation is already exact

Correct Answer: An integrating factor μ(x) exists

Q18. If (∂N/∂x – ∂M/∂y)/M is a function of y only, then the suitable integrating factor is:

  • μ(x) = exp(∫[(∂M/∂y – ∂N/∂x)/N] dx)
  • μ(y) = exp(∫[(∂N/∂x – ∂M/∂y)/M] dy)
  • μ(x,y) = x+y
  • No integrating factor can be found

Correct Answer: μ(y) = exp(∫[(∂N/∂x – ∂M/∂y)/M] dy)

Q19. A necessary step after multiplying by an integrating factor is:

  • Re-check exactness with new M and N
  • Differentiate the integrating factor again
  • Discard original equation
  • Assume solution is linear

Correct Answer: Re-check exactness with new M and N

Q20. Which of the following equations is exact?

  • (y – 2x)dx + (x + 3y)dy = 0
  • (2xy)dx + (x^2)dy = 0
  • (e^x y)dx + (e^x)dy = 0
  • ((sin x)dx + (cos y)dy = 0)

Correct Answer: (e^x y)dx + (e^x)dy = 0

Q21. Solve the exact equation e^x(y + 1)dx + e^x dy = 0. The implicit solution φ(x,y) = C is:

  • e^x y + e^x + C
  • e^x y + C
  • e^x (y + 1) + C
  • e^x y + e^x + C = 0

Correct Answer: e^x y + e^x + C

Q22. In constructing φ(x,y), if φ_x = M, integrating M with respect to x yields:

  • φ(x,y) + constant
  • ∫M dx + g(y)
  • ∫M dy + h(x)
  • M + N

Correct Answer: ∫M dx + g(y)

Q23. For the equation (y + 2x)dx + x dy = 0, check exactness: ∂M/∂y = 1 and ∂N/∂x = 1. The general solution is:

  • φ = xy + x^2 + C
  • φ = y + x^2 + C
  • φ = x + y + C
  • φ = x^2 + C

Correct Answer: φ = xy + x^2 + C

Q24. Which statement about integrating factors is correct?

  • Integrating factors always depend on both x and y
  • Integrating factors can depend only on x or only on y
  • Integrating factors are always polynomials
  • If an equation is non-exact, no integrating factor exists

Correct Answer: Integrating factors can depend only on x or only on y

Q25. Given M = (2xy + y^2) and N = x^2 + 2xy, an integrating factor μ(x) depending only on x exists because:

  • (∂M/∂y – ∂N/∂x)/N is function of x only
  • (∂N/∂x – ∂M/∂y)/M is function of y only
  • Both M and N are exact already
  • No integrating factor exists

Correct Answer: Both M and N are exact already

Q26. For the differential equation (x^2 y – y)dx + (x^3 + x)dy = 0, compute ∂M/∂y = x^2 – 1 and ∂N/∂x = 3x^2 + 1. Is it exact?

  • Yes, because derivatives match
  • No, because ∂M/∂y ≠ ∂N/∂x
  • Yes, because M and N are polynomials
  • No, because degree differs

Correct Answer: No, because ∂M/∂y ≠ ∂N/∂x

Q27. In many pharmaceutical applications, exact equations help ensure:

  • Mass balance or conservation laws are satisfied
  • Manufacturing processes are faster
  • Tablets dissolve instantly
  • Drug discovery is automated

Correct Answer: Mass balance or conservation laws are satisfied

Q28. If an integrating factor μ(x) = x^k makes Mdx + Ndy exact, the process involves:

  • Multiplying M and N by x^k and checking ∂(μM)/∂y = ∂(μN)/∂x
  • Adding x^k to M and N
  • Dividing M and N by x^k
  • Integrating x^k separately

Correct Answer: Multiplying M and N by x^k and checking ∂(μM)/∂y = ∂(μN)/∂x

Q29. For M = y/(x^2 + y^2) and N = -x/(x^2 + y^2), is Mdx + Ndy exact in region excluding origin?

  • Yes, because ∂M/∂y = ∂N/∂x everywhere
  • No, because ∂M/∂y ≠ ∂N/∂x
  • Yes, but only when x>0
  • No, because singularity at origin prevents any exactness

Correct Answer: No, because ∂M/∂y ≠ ∂N/∂x

Q30. Which method is appropriate for solving exact differential equations arising in compartmental PK models?

  • Construct potential function φ and set φ = C to represent conserved quantity
  • Use numerical integration only
  • Ignore coupling between compartments
  • Apply Fourier transforms

Correct Answer: Construct potential function φ and set φ = C to represent conserved quantity

Q31. For the equation (3x^2 y + sin y)dx + (x^3 + cos y)dy = 0, check exactness: ∂M/∂y = 3x^2 + cos y, ∂N/∂x = 3x^2. Is it exact?

  • Yes, exact everywhere
  • No, not exact because ∂M/∂y ≠ ∂N/∂x
  • Yes, if y is small
  • No, because cos y is zero

Correct Answer: No, not exact because ∂M/∂y ≠ ∂N/∂x

Q32. If an integrating factor μ depends on function of x and y through combination xy, then μ = μ(xy) implies:

  • Transform variable to u = xy may simplify finding μ
  • μ must be constant
  • μ depends only on x
  • μ depends only on y

Correct Answer: Transform variable to u = xy may simplify finding μ

Q33. The general approach to find φ after confirming exactness is:

  • Integrate N with respect to x
  • Integrate M with respect to x, then determine g(y) and enforce φ_y = N
  • Differentiate M with respect to x and integrate again
  • Set M equal to N

Correct Answer: Integrate M with respect to x, then determine g(y) and enforce φ_y = N

Q34. Which of the following is true about first-order exact ODEs?

  • They always have unique explicit y(x) solutions
  • The implicit solution φ(x,y) = C often suffices for B.Pharm applications
  • They cannot be solved analytically
  • They always require numerical methods

Correct Answer: The implicit solution φ(x,y) = C often suffices for B.Pharm applications

Q35. For M = 2xy and N = x^2, ∂M/∂y = 2x and ∂N/∂x = 2x. The implicit solution φ is:

  • φ = x^2 y + C
  • φ = x y + C
  • φ = x^2 + y + C
  • φ = 2x y + C

Correct Answer: φ = x^2 y + C

Q36. If a differential equation is exact on a simply connected domain, the potential function φ exists and is:

  • Unique up to an additive constant
  • Completely arbitrary
  • Dependent on boundary conditions only
  • Non-differentiable

Correct Answer: Unique up to an additive constant

Q37. An integrating factor depending only on y for equation Mdx + Ndy = 0 can be found when:

  • (∂M/∂y – ∂N/∂x)/N is function of x only
  • (∂N/∂x – ∂M/∂y)/M is function of y only
  • ∂M/∂x = ∂N/∂y
  • Both M and N are independent of y

Correct Answer: (∂N/∂x – ∂M/∂y)/M is function of y only

Q38. For M = y e^{x} and N = e^{x}(y + 1), is the equation exact?

  • No, because derivatives differ
  • Yes, because ∂M/∂y = e^{x} and ∂N/∂x = e^{x}(y + 1)
  • No, because e^{x} ruins exactness
  • Yes, because ∂M/∂y = e^{x} and ∂N/∂x = e^{x}

Correct Answer: No, because derivatives differ

Q39. When solving exact equations numerically for pharmacokinetic models, the implicit φ(x,y)=C is useful for:

  • Verifying conservation or invariants
  • Direct calculation of absorption rates
  • Replacing numerical solvers entirely
  • None of the above

Correct Answer: Verifying conservation or invariants

Q40. Which of the following is a correct integrating factor for (y – x)dx + (x – y)dy = 0?

  • μ(x) = e^x
  • μ(y) = e^y
  • μ(x,y) = 1/(x – y)
  • No simple integrating factor depending only on x or y exists

Correct Answer: No simple integrating factor depending only on x or y exists

Q41. If φ(x,y) is a potential of exact equation, then total differential dφ equals:

  • M dx – N dy
  • M dx + N dy
  • N dx + M dy
  • 0

Correct Answer: M dx + N dy

Q42. Consider differential equation (2xy – y^2)dx + (x^2 – 2xy)dy = 0. Check exactness and find the nature:

  • Exact because ∂M/∂y = ∂N/∂x
  • Not exact because derivatives differ
  • Exact because both M and N are homogeneous of degree 2
  • Linear not exact

Correct Answer: Exact because ∂M/∂y = ∂N/∂x

Q43. For a potential φ found by integrating M with respect to x, the next step is to:

  • Integrate φ with respect to x again
  • Differentiate the result with respect to y and compare to N
  • Divide by N
  • Solve for x explicitly

Correct Answer: Differentiate the result with respect to y and compare to N

Q44. The integrating factor μ(x) = 1/x for equation (y/x)dx + (ln x)dy = 0 would be used when:

  • Equation is already exact
  • Multiplying by 1/x yields exactness
  • Equation is separable only
  • It simplifies to linear ODE in y

Correct Answer: Multiplying by 1/x yields exactness

Q45. For M(x,y)=sin(y) and N(x,y)=x cos(y), check exactness: ∂M/∂y=cos(y), ∂N/∂x=cos(y). What is φ?

  • φ = x sin(y) + C
  • φ = cos(y) + C
  • φ = x sin(y) – y + C
  • φ = sin(y) + x + C

Correct Answer: φ = x sin(y) + C

Q46. Which integration order is commonly used when M has simpler antiderivative in x?

  • Integrate N with respect to y first
  • Integrate M with respect to x first
  • Differentiate both M and N first
  • Use Laplace transforms

Correct Answer: Integrate M with respect to x first

Q47. In the context of drug release, an exact differential might represent:

  • A balance between release and absorption rates leading to conservation
  • Only the dissolution rate independent of absorption
  • An algebraic relation not involving time
  • A thermodynamic constant

Correct Answer: A balance between release and absorption rates leading to conservation

Q48. For equation (x dy – y dx)/(x^2) = 0, rewriting gives (-y/x^2)dx + (1/x)dy = 0. Is it exact?

  • Yes, because ∂M/∂y = ∂N/∂x
  • No, because ∂M/∂y ≠ ∂N/∂x
  • Yes, after multiplying by x
  • No, because division by x^2 is invalid

Correct Answer: Yes, after multiplying by x

Q49. If M and N are continuous with continuous partials in a simply connected domain and ∂M/∂y = ∂N/∂x, then:

  • No solution exists
  • There exists a scalar potential φ with dφ = M dx + N dy
  • Equation must be linear
  • Integrating factor is required

Correct Answer: There exists a scalar potential φ with dφ = M dx + N dy

Q50. Which best summarizes solving exact differential equations for B.Pharm students?

  • Always use numerical methods for pharmacology problems
  • Check exactness, find integrating factor if needed, construct φ, and interpret the implicit solution in context
  • Ignore exactness and apply separation of variables
  • Solve only linear ODEs as exact equations

Correct Answer: Check exactness, find integrating factor if needed, construct φ, and interpret the implicit solution in context

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