BMath Bachelor of Mathematics Differential Equations 1 — Questions and Answers
Question 1: What is the order of the differential equation d²y/dx² + 3 dy/dx + 2y = 0?
- 1
- 2 (Correct answer)
- 3
- 0
Correct answer: 2
The order of a differential equation is the highest derivative present; here it is d²y/dx², which is second order.
Question 2: The general solution of dy/dx = ky is:
- y = kx + C
- y = Ce^(kx) (Correct answer)
- y = Cx^k
- y = k ln x + C
Correct answer: y = Ce^(kx)
Separating variables: dy/y = k dx, integrating gives ln|y| = kx + C₀, so y = Ce^(kx).
Question 3: Which method is used to find a particular solution of a nonhomogeneous linear ODE?
- Separation of variables
- Method of undetermined coefficients (Correct answer)
- Euler's method
- Wronskian method
Correct answer: Method of undetermined coefficients
The method of undetermined coefficients guesses a particular solution form matching the forcing function and solves for the coefficients.
Question 4: For the IVP dy/dx = f(x, y), y(x₀) = y₀, Euler's method approximates y(x₀ + h) as:
- y₀ + h·f(x₀, y₀) (Correct answer)
- y₀ − h·f(x₀, y₀)
- y₀·f(x₀, y₀)
- y₀ + h²·f(x₀, y₀)
Correct answer: y₀ + h·f(x₀, y₀)
Euler's method uses the tangent line approximation: y₁ ≈ y₀ + h·f(x₀, y₀), stepping forward by step size h.
Question 5: The Wronskian W(y₁, y₂) of two solutions is used to determine:
- Whether the solutions are linearly independent (Correct answer)
- The particular solution of the ODE
- The eigenvalues of the system
- The order of the ODE
Correct answer: Whether the solutions are linearly independent
If W(y₁, y₂) ≠ 0 on an interval, then y₁ and y₂ are linearly independent and form a fundamental set of solutions.
Question 6: The characteristic equation of y'' − 5y' + 6y = 0 is:
- r² − 5r + 6 = 0 (Correct answer)
- r² + 5r − 6 = 0
- r² + 5r + 6 = 0
- r − 5r² + 6 = 0
Correct answer: r² − 5r + 6 = 0
Substituting y = e^(rx) into the ODE yields the characteristic equation r² − 5r + 6 = 0, with roots r = 2 and r = 3.
BMath Bachelor of Mathematics Differential Equations 1
What is the order of the differential equation d²y/dx² + 3 dy/dx + 2y = 0?