Homogeneous differential equations MCQs With Answer are essential for B.Pharm students studying pharmacokinetics and mathematical modeling. This collection focuses on first-order homogeneous equations reducible by substitution (y = vx), linear homogeneous ODEs with constant coefficients, Euler–Cauchy equations, and homogeneous systems used in compartment models. Questions link core theory—characteristic equations, eigenvalues, and general solution forms—to practical pharmacy applications like drug elimination, steady-state behavior, and stability analysis. Each MCQ reinforces problem-solving steps, interpretation of solutions, and exam-ready techniques tailored to B.Pharm curricula. Now let’s test your knowledge with 50 MCQs on this topic.
Q1. Which of the following first-order equations is homogeneous (f(tx,ty)=f(x,y))?
- dy/dx = (x + y) / x
- dy/dx = (x + 2) / y
- dy/dx = (x + y) / (x – y)
- dy/dx = x + y
Correct Answer: dy/dx = (x + y) / (x – y)
Q2. For the homogeneous first-order DE dy/dx = (x – y)/(x + y), which substitution reduces it to a separable equation?
- y = ux + 1
- y = vx
- x = vy
- y = e^{vx}
Correct Answer: y = vx
Q3. A linear homogeneous ODE with constant coefficients has which general solution structure?
- Sum of particular solutions only
- Sum of exponentials formed from roots of the characteristic equation
- Only polynomial solutions
- Only trigonometric solutions
Correct Answer: Sum of exponentials formed from roots of the characteristic equation
Q4. The ODE y” – 4y’ + 5y = 0 has characteristic roots r = 2 ± i. What is the general real solution?
- y = C1 e^{2x} + C2 e^{-2x}
- y = e^{2x}(C1 cos x + C2 sin x)
- y = C1 cos 2x + C2 sin 2x
- y = C1 e^{x} + C2 e^{5x}
Correct Answer: y = e^{2x}(C1 cos x + C2 sin x)
Q5. In pharmacokinetics, the one-compartment first-order elimination model dy/dt = -k y is a homogeneous linear ODE. What is the solution for y(t) given y(0)=y0?
- y(t) = y0 + kt
- y(t) = y0 e^{-kt}
- y(t) = k t + y0 e^{kt}
- y(t) = y0 / (1 + kt)
Correct Answer: y(t) = y0 e^{-kt}
Q6. The half-life for a first-order elimination process is given by which formula?
- t1/2 = ln(2) / k
- t1/2 = k / ln(2)
- t1/2 = 2 / k
- t1/2 = ln(k) / 2
Correct Answer: t1/2 = ln(2) / k
Q7. Which property characterizes a homogeneous function f(x,y) of degree n?
- f(x,y) = f(y,x) for all x,y
- f(tx,ty) = t^n f(x,y) for all t
- f(x,y) is linear in x and y
- f(x,y) has only even powers of x and y
Correct Answer: f(tx,ty) = t^n f(x,y) for all t
Q8. The equation x^2 y” + 3x y’ + y = 0 is an Euler–Cauchy equation. Which substitution is standard to solve it?
- y = e^{mx}
- x = e^t, then set y(x)=Y(t)
- y = vx
- Use Laplace transform directly
Correct Answer: x = e^t, then set y(x)=Y(t)
Q9. For the ODE y” + 4y = 0, which pair of functions forms a fundamental set of solutions?
- e^{2x} and e^{-2x}
- cos 2x and sin 2x
- e^{4x} and x e^{4x}
- 1 and x
Correct Answer: cos 2x and sin 2x
Q10. When the characteristic equation has a repeated root r of multiplicity m, the corresponding independent solutions include which forms?
- e^{rt}, t e^{rt}, t^2 e^{rt}, …, t^{m-1} e^{rt}
- only e^{rt} repeated m times
- sin rt and cos rt repeated
- polynomials of degree m only
Correct Answer: e^{rt}, t e^{rt}, t^2 e^{rt}, …, t^{m-1} e^{rt}
Q11. Which statement about homogeneous linear ODEs is true?
- The zero function is never a solution
- Any constant multiple of a solution is also a solution
- Sum of two solutions is not a solution
- They always have no real solutions
Correct Answer: Any constant multiple of a solution is also a solution
Q12. The Wronskian of two solutions y1 and y2 of a second-order linear homogeneous ODE is zero for all x. What does this imply?
- y1 and y2 are linearly independent
- y1 and y2 are linearly dependent
- The ODE has no solutions
- The ODE is non-homogeneous
Correct Answer: y1 and y2 are linearly dependent
Q13. The system for a two-compartment model without input can be written as dx/dt = Ax. This is a homogeneous linear system. How are solutions commonly expressed?
- x(t) = A t
- x(t) = e^{At} x(0)
- x(t) = x(0) + At
- x(t) = \int_0^t A dt
Correct Answer: x(t) = e^{At} x(0)
Q14. Which method is appropriate to solve the homogeneous first-order equation dy/dx = F(y/x)?
- Substitute x = v y
- Substitute y = vx
- Use integrating factor
- Use variation of parameters
Correct Answer: Substitute y = vx
Q15. The homogeneous ODE y’ = (x^2 + xy)/(x^2) simplifies after dividing numerator and denominator by x^2 to what function of y/x?
- 1 + y/x
- 1 + x/y
- y/x + x/y
- only y
Correct Answer: 1 + y/x
Q16. For y” – y = 0, which functions are solutions?
- e^{x} and e^{-x}
- cosh x and sinh x
- Both of the above
- None of the above
Correct Answer: Both of the above
Q17. In a homogeneous linear system dx/dt = Ax, stability of the equilibrium x=0 depends on which property?
- Determinant of A only
- Eigenvalues of A
- Trace of A only
- Rank of A only
Correct Answer: Eigenvalues of A
Q18. The equation dy/dx = (x^2 + y^2)/(xy) is homogeneous. After substitution y = vx, the resulting ODE in v and x is which type?
- Linear in v
- Separable in v and x
- Exact but not separable
- Non-reducible
Correct Answer: Separable in v and x
Q19. Which of the following is a homogeneous linear differential operator of order 3?
- L[y] = y”’ + p(x) y” + q(x) y’ + r(x) y
- L[y] = y”’ + p(x) y” + q(x) y’ + r(x)
- L[y] = y”’ + p(x)
- L[y] = y”’ + p(x) y + q(x)
Correct Answer: L[y] = y”’ + p(x) y” + q(x) y’ + r(x) y
Q20. Which equation is non-homogeneous?
- y” + 2y’ + y = 0
- y” + y = sin x
- x^2 y” – x y’ + y = 0
- y’ + p(x) y = 0
Correct Answer: y” + y = sin x
Q21. The principle of superposition applies to which equations?
- Linear homogeneous differential equations
- Nonlinear equations only
- Any differential equation
- Linear non-homogeneous equations without forcing
Correct Answer: Linear homogeneous differential equations
Q22. For dy/dx = (ax + by)/(cx + dy) to be homogeneous of degree 0, which condition must hold?
- a = c and b = d
- The numerator and denominator must be homogeneous polynomials of same degree
- All coefficients must be zero
- It must be linear in x only
Correct Answer: The numerator and denominator must be homogeneous polynomials of same degree
Q23. Which substitution turns the homogeneous equation y’ = f(y/x) into a separable equation?
- x = uy
- y = vx
- y = e^{u}
- x = e^{u}
Correct Answer: y = vx
Q24. In solving homogeneous linear ODEs with constant coefficients, complex conjugate roots α ± iβ produce solutions involving which functions?
- e^{αx} cos βx and e^{αx} sin βx
- e^{βx} cos αx
- Only cos αx and sin αx
- Polynomials times e^{αx}
Correct Answer: e^{αx} cos βx and e^{αx} sin βx
Q25. The homogeneous second-order ODE y” + p(x)y’ + q(x) y = 0 has two solutions y1 and y2. If W(y1,y2) ≠ 0 at some point, then:
- y1 and y2 are linearly independent on an interval
- y1 and y2 must be identical
- The equation has no other solutions
- The equation is non-homogeneous
Correct Answer: y1 and y2 are linearly independent on an interval
Q26. For the homogeneous first-order DE (x+y) dx – (x-y) dy = 0, which approach is best?
- Recognize exact equation and integrate
- Divide by x and use y = vx substitution
- Differentiate both sides directly
- Apply Laplace transform
Correct Answer: Divide by x and use y = vx substitution
Q27. Which describes an equidimensional (Euler–Cauchy) ODE?
- Coefficients are powers of x matching derivative order
- Coefficients are trigonometric
- It always has constant coefficients
- It cannot be solved analytically
Correct Answer: Coefficients are powers of x matching derivative order
Q28. A homogeneous linear ODE y” + a y’ + b y = 0 with a^2 – 4b < 0 yields what behavior?
- Overdamped exponential decay
- Underdamped oscillatory behavior
- Critical damping
- No real solution
Correct Answer: Underdamped oscillatory behavior
Q29. In pharmacokinetic modeling, the homogeneous system dx/dt = Ax with negative real parts of eigenvalues implies:
- Unbounded growth of drug concentration
- Drug concentrations decay to zero over time
- Oscillatory increase indefinitely
- No change in concentrations
Correct Answer: Drug concentrations decay to zero over time
Q30. The differential equation dy/dx = (x^2 + xy + y^2)/x^2 is homogeneous. After y = vx substitution, the ODE becomes dv/dx = g(v)/x. This indicates solutions often involve which operation?
- Integration of g(v) dv equals ln x plus constant
- Direct exponentiation with x
- Solving algebraically for v
- Applying Fourier transform
Correct Answer: Integration of g(v) dv equals ln x plus constant
Q31. Which of the following ODEs is homogeneous of degree 1 in x and y?
- dy/dx = (x^2 + y)/x
- dy/dx = (x + y)/(x + 2y)
- dy/dx = (x^2 + y^2)/(xy)
- dy/dx = y/x + 1/x
Correct Answer: dy/dx = (x + y)/(x + 2y)
Q32. The homogeneous linear ODE y” + 6y’ + 9y = 0 has a repeated root r = -3. What is the general solution?
- y = C1 e^{-3x} + C2 e^{3x}
- y = (C1 + C2 x) e^{-3x}
- y = C1 cos 3x + C2 sin 3x
- y = C1 e^{-9x}
Correct Answer: y = (C1 + C2 x) e^{-3x}
Q33. Bernoulli’s equation y’ + P(x) y = Q(x) y^n is nonlinear but can be made linear by which substitution?
- z = y^{1-n}
- z = ln y
- z = y^n
- z = 1/y
Correct Answer: z = y^{1-n}
Q34. If y1 and y2 are solutions of a homogeneous second-order ODE, then for constants c1 and c2, c1 y1 + c2 y2 is:
- Always a solution
- Never a solution
- A solution only if c1 = c2
- A solution only if y1 = y2
Correct Answer: Always a solution
Q35. Which is the complementary function (CF) for y” – y’ – 2y = 0?
- y_c = C1 e^{2x} + C2 e^{-x}
- y_c = C1 e^{x} + C2 e^{-2x}
- y_c = C1 cos x + C2 sin x
- y_c = C1 + C2 x
Correct Answer: y_c = C1 e^{2x} + C2 e^{-x}
Q36. To check if a first-order ODE M(x,y) dx + N(x,y) dy = 0 is homogeneous, you can test whether:
- M and N are both homogeneous functions of same degree
- M and N have constant values
- M depends only on x and N only on y
- Equation is exact
Correct Answer: M and N are both homogeneous functions of same degree
Q37. Which technique is NOT typically used for homogeneous linear ODEs?
- Characteristic equation method
- Reduction of order
- Variation of parameters for homogeneous case
- Method of undetermined coefficients to find complementary function
Correct Answer: Method of undetermined coefficients to find complementary function
Q38. For the homogeneous system dx/dt = Ax, if A has distinct real eigenvalues λ1 and λ2 with eigenvectors v1 and v2, the general solution is:
- x(t) = C1 v1 e^{λ1 t} + C2 v2 e^{λ2 t}
- x(t) = (v1 + v2) e^{At}
- x(t) = e^{A(t+1)}
- x(t) = A^{-1} e^{t} x(0)
Correct Answer: x(t) = C1 v1 e^{λ1 t} + C2 v2 e^{λ2 t}
Q39. Which of the following is a homogeneous linear ODE of order one?
- y’ + p(x) y = g(x)
- y’ + p(x) y = 0
- y” + p(x) y’ + q(x) y = r(x)
- y” = f(x,y)
Correct Answer: y’ + p(x) y = 0
Q40. In solving homogeneous first-order ODEs, after substituting y = vx and simplifying, you often integrate to find v as a function of x. The integration constant typically appears as:
- C multiplied by x
- C added to v
- ln x + C or x^C depending on separation
- No constant appears for homogeneous equations
Correct Answer: ln x + C or x^C depending on separation
Q41. For the homogeneous ODE dy/dx = (3x + y)/(x – y), the substitution y = vx yields an ODE in v. What is the derivative dy/dx in terms of v and dv/dx?
- dy/dx = v + x dv/dx
- dy/dx = x + v dv/dx
- dy/dx = v x dv/dx
- dy/dx = dv/dx
Correct Answer: dy/dx = v + x dv/dx
Q42. The homogeneous linear equation y” + ω^2 y = 0 models which pharmacological behavior analogously?
- Exponential drug elimination
- Simple harmonic oscillation (theoretical oscillatory concentration in idealized models)
- Zero-order elimination
- Constant infusion with steady increase
Correct Answer: Simple harmonic oscillation (theoretical oscillatory concentration in idealized models)
Q43. Which statement accurately distinguishes homogeneous vs non-homogeneous linear ODEs?
- Homogeneous ODEs have zero on one side; non-homogeneous have nonzero forcing term
- Homogeneous ODEs are nonlinear
- Non-homogeneous ODEs always have no solution
- They are the same concept
Correct Answer: Homogeneous ODEs have zero on one side; non-homogeneous have nonzero forcing term
Q44. For a linear homogeneous ODE, the space of solutions forms which mathematical structure?
- A vector space
- A ring
- A metric space only
- A probability space
Correct Answer: A vector space
Q45. If two solutions of a homogeneous second-order ODE are known, the general solution can be constructed by:
- Multiplying them together
- Taking linear combinations with constants
- Dividing one by the other
- Integrating their product
Correct Answer: Taking linear combinations with constants
Q46. The ODE dy/dx = (x^3 + x^2 y)/(x^3) simplifies to which homogeneous function of y/x?
- 1 + y/x
- 1 + (y/x)/x
- (x^2 + y)/x
- Only x^2
Correct Answer: 1 + y/x
Q47. For the homogeneous linear ODE y” + p y’ + q y = 0, how many arbitrary constants will the general solution contain?
- One
- Two
- Three
- Zero
Correct Answer: Two
Q48. In a B.Pharm exam, recognizing that an ODE is homogeneous allows you to:
- Immediately write the solution without work
- Apply substitution y = vx to simplify and separate variables
- Conclude the equation is unsolvable
- Use numerical methods only
Correct Answer: Apply substitution y = vx to simplify and separate variables
Q49. A homogeneous differential equation of degree zero implies the dependent variable scaling has what effect?
- Scaling x and y by t multiplies f by t
- Scaling (tx,ty) leaves the ratio y/x unchanged so f(tx,ty)=f(x,y)
- It makes the equation linear in x only
- Dependent variable becomes constant
Correct Answer: Scaling (tx,ty) leaves the ratio y/x unchanged so f(tx,ty)=f(x,y)
Q50. For the homogeneous linear ODE y” – 2y’ + y = 0, which initial conditions y(0)=1, y'(0)=0 yield which particular solution?
- y = e^{x}
- y = e^{x}(1 – x)
- y = e^{x}(1 + x)
- y = e^{x} x
Correct Answer: y = e^{x}(1 – x)
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