Mathematics ยท Physics

Differential Equations

190 Questions

Differential equations involve finding functions that relate variables to their rates of change. These questions cover first and second order equations and separation of variables. They are a core part of the mathematics syllabus for high level competitive exams.

First order separationSecond order linear equationsInitial value problemsPredator prey modelsCauchy Riemann equations

Differential Equations Questions

Multiple choice

Consider the system of differential equations (\frac{dx}{dt} = 3x + 2y, \ \frac{dy}{dt} = -x + y). Determine the type of equilibrium point at the origin.

  1. Saddle point

  2. Center

  3. Stable node

  4. Unstable node
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To determine the type of equilibrium point, we find the eigenvalues of the coefficient matrix. The eigenvalues are (\lambda_1 = 2) and (\lambda_2 = -1). Since the eigenvalues have opposite signs, the equilibrium point at the origin is a saddle point.

Multiple choice

Given the system of differential equations (\frac{dx}{dt} = -2x + y, \ \frac{dy}{dt} = -x - 2y), find the solution that satisfies the initial conditions (x(0) = 1, y(0) = 2).

  1. \(x(t) = e^{-3t} (\cos t + \sin t), \\ y(t) = e^{-3t} (\cos t - \sin t)\)
  2. \(x(t) = e^{-t} (\cos 2t + \sin 2t), \\ y(t) = e^{-t} (\cos 2t - \sin 2t)\)
  3. \(x(t) = e^{-5t} (\cos t + \sin t), \\ y(t) = e^{-5t} (\cos t - \sin t)\)
  4. \(x(t) = e^{-t} (\cos t - \sin t), \\ y(t) = e^{-t} (\cos t + \sin t)\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To find the solution, we first find the eigenvalues and eigenvectors of the coefficient matrix. Then, we construct the fundamental matrix and use it to solve the system of differential equations with the given initial conditions.

Multiple choice

Consider the system of differential equations (\frac{dx}{dt} = 2x - 3y, \ \frac{dy}{dt} = x + 2y). Find the eigenvalues of the coefficient matrix.

  1. \(\lambda_1 = 1, \lambda_2 = 2\)
  2. \(\lambda_1 = -1, \lambda_2 = -2\)
  3. \(\lambda_1 = 3, \lambda_2 = 4\)
  4. \(\lambda_1 = -3, \lambda_2 = -4\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The eigenvalues are found by solving the characteristic equation (\det(A - \lambda I) = 0), where (A) is the coefficient matrix. Solving this equation gives ((\lambda - 1)(\lambda - 2) = 0), which has eigenvalues (\lambda_1 = 1) and (\lambda_2 = 2).

Multiple choice

Given the system of differential equations (\frac{dx}{dt} = -x + 2y, \ \frac{dy}{dt} = -2x - y), find the general solution using the Laplace transform method.

  1. \(x(t) = e^{-t} (\cos t + \sin t), \\ y(t) = e^{-t} (\cos t - \sin t)\)
  2. \(x(t) = e^{-2t} (\cos 2t + \sin 2t), \\ y(t) = e^{-2t} (\cos 2t - \sin 2t)\)
  3. \(x(t) = e^{-3t} (\cos t + \sin t), \\ y(t) = e^{-3t} (\cos t - \sin t)\)
  4. \(x(t) = e^{-t} (\cos t - \sin t), \\ y(t) = e^{-t} (\cos t + \sin t)\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the Laplace transform method, we first transform the system of differential equations into a system of algebraic equations. Then, we solve the algebraic equations and apply the inverse Laplace transform to obtain the general solution.

Multiple choice

Consider the system of differential equations (\frac{dx}{dt} = 4x - 2y, \ \frac{dy}{dt} = 2x + y). Determine the stability of the equilibrium point at the origin.

  1. Stable

  2. Unstable

  3. Center

  4. Semi-stable
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To determine the stability, we find the eigenvalues of the coefficient matrix. The eigenvalues are (\lambda_1 = 3 + i\sqrt{7}) and (\lambda_2 = 3 - i\sqrt{7}). Since both eigenvalues have negative real parts, the equilibrium point at the origin is stable.

Multiple choice

Given the system of differential equations (\frac{dx}{dt} = x + y, \ \frac{dy}{dt} = -x + y), find the solution that satisfies the initial conditions (x(0) = 2, y(0) = 1).

  1. \(x(t) = e^t (\cos t + \sin t), \\ y(t) = e^t (\cos t - \sin t)\)
  2. \(x(t) = e^{-t} (\cos t + \sin t), \\ y(t) = e^{-t} (\cos t - \sin t)\)
  3. \(x(t) = e^{2t} (\cos t + \sin t), \\ y(t) = e^{2t} (\cos t - \sin t)\)
  4. \(x(t) = e^{-2t} (\cos t + \sin t), \\ y(t) = e^{-2t} (\cos t - \sin t)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the solution, we first find the eigenvalues and eigenvectors of the coefficient matrix. Then, we construct the fundamental matrix and use it to solve the system of differential equations with the given initial conditions.

Multiple choice

Consider the system of differential equations (\frac{dx}{dt} = 2x + 3y, \ \frac{dy}{dt} = -x + 2y). Find the eigenvalues of the coefficient matrix.

  1. \(\lambda_1 = 1, \lambda_2 = 2\)
  2. \(\lambda_1 = -1, \lambda_2 = -2\)
  3. \(\lambda_1 = 3, \lambda_2 = 4\)
  4. \(\lambda_1 = -3, \lambda_2 = -4\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The eigenvalues are found by solving the characteristic equation (\det(A - \lambda I) = 0), where (A) is the coefficient matrix. Solving this equation gives ((\lambda - 1)(\lambda - 2) = 0), which has eigenvalues (\lambda_1 = 1) and (\lambda_2 = 2).

Multiple choice

Given the system of differential equations (\frac{dx}{dt} = -3x + 2y, \ \frac{dy}{dt} = 2x + 3y), find the general solution using the method of undetermined coefficients.

  1. \(x(t) = e^{-t} (\cos t + \sin t), \\ y(t) = e^{-t} (\cos t - \sin t)\)
  2. \(x(t) = e^{-2t} (\cos 2t + \sin 2t), \\ y(t) = e^{-2t} (\cos 2t - \sin 2t)\)
  3. \(x(t) = e^{-3t} (\cos t + \sin t), \\ y(t) = e^{-3t} (\cos t - \sin t)\)
  4. \(x(t) = e^{-t} (\cos t - \sin t), \\ y(t) = e^{-t} (\cos t + \sin t)\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the method of undetermined coefficients, we first guess the form of the solution. Then, we substitute the guess into the system of differential equations and solve for the coefficients.

Multiple choice

Consider the system of differential equations (\frac{dx}{dt} = -4x + 3y, \ \frac{dy}{dt} = 3x + 4y). Determine the stability of the equilibrium point at the origin.

  1. Stable

  2. Unstable

  3. Center

  4. Semi-stable
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To determine the stability, we find the eigenvalues of the coefficient matrix. The eigenvalues are (\lambda_1 = -1 + i\sqrt{7}) and (\lambda_2 = -1 - i\sqrt{7}). Since both eigenvalues have negative real parts, the equilibrium point at the origin is stable.

Multiple choice

Given the system of differential equations (\frac{dx}{dt} = x - 2y, \ \frac{dy}{dt} = 2x + y), find the solution that satisfies the initial conditions (x(0) = 1, y(0) = 2).

  1. \(x(t) = e^t (\cos t + \sin t), \\ y(t) = e^t (\cos t - \sin t)\)
  2. \(x(t) = e^{-t} (\cos t + \sin t), \\ y(t) = e^{-t} (\cos t - \sin t)\)
  3. \(x(t) = e^{2t} (\cos t + \sin t), \\ y(t) = e^{2t} (\cos t - \sin t)\)
  4. \(x(t) = e^{-2t} (\cos t + \sin t), \\ y(t) = e^{-2t} (\cos t - \sin t)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the solution, we first find the eigenvalues and eigenvectors of the coefficient matrix. Then, we construct the fundamental matrix and use it to solve the system of differential equations with the given initial conditions.