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 differential equation $\frac{dy}{dx} = \frac{x^2}{y^2}$. Which of the following statements is true?

  1. The equation has a unique solution for any initial condition.

  2. The equation has a unique solution for some initial conditions.

  3. The equation has no solutions.

  4. The equation has infinitely many solutions.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is not Lipschitz continuous in $y$, so it may not have a unique solution for any initial condition. In fact, the equation has no solutions, as can be seen by separation of variables.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{y^2}{x^2}$. Which of the following statements is true?

  1. The equation has a unique solution for any initial condition.

  2. The equation has a unique solution for some initial conditions.

  3. The equation has no solutions.

  4. The equation has infinitely many solutions.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is not Lipschitz continuous in $y$, so it may not have a unique solution for any initial condition. In fact, the equation has no solutions, as can be seen by separation of variables.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{x^3}{y^3}$. Which of the following statements is true?

  1. The equation has a unique solution for any initial condition.

  2. The equation has a unique solution for some initial conditions.

  3. The equation has no solutions.

  4. The equation has infinitely many solutions.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is not Lipschitz continuous in $y$, so it may not have a unique solution for any initial condition. In fact, the equation has no solutions, as can be seen by separation of variables.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{y^3}{x^3}$. Which of the following statements is true?

  1. The equation has a unique solution for any initial condition.

  2. The equation has a unique solution for some initial conditions.

  3. The equation has no solutions.

  4. The equation has infinitely many solutions.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is not Lipschitz continuous in $y$, so it may not have a unique solution for any initial condition. In fact, the equation has no solutions, as can be seen by separation of variables.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{x^4}{y^4}$. Which of the following statements is true?

  1. The equation has a unique solution for any initial condition.

  2. The equation has a unique solution for some initial conditions.

  3. The equation has no solutions.

  4. The equation has infinitely many solutions.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is not Lipschitz continuous in $y$, so it may not have a unique solution for any initial condition. In fact, the equation has no solutions, as can be seen by separation of variables.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{y^4}{x^4}$. Which of the following statements is true?

  1. The equation has a unique solution for any initial condition.

  2. The equation has a unique solution for some initial conditions.

  3. The equation has no solutions.

  4. The equation has infinitely many solutions.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is not Lipschitz continuous in $y$, so it may not have a unique solution for any initial condition. In fact, the equation has no solutions, as can be seen by separation of variables.

Multiple choice

Consider the differential equation $\frac{dy}{dx} + y = x$. Which of the following is the integrating factor for this equation?

  1. $e^{-x}$
  2. $e^x$
  3. $x$
  4. $1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The integrating factor for the given equation is $e^x$ because it makes the equation exact.

Multiple choice

Which of the following is a first order linear differential equation?

  1. $y' + y = x^2$
  2. $y'' + 2y' + y = 0$
  3. $y' = x^2 + y^2$
  4. $y''' - 3y'' + 2y' - y = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A first order linear differential equation is one in which the dependent variable and its derivatives appear linearly.

Multiple choice

Use the method of separation of variables to solve the following differential equation: $\frac{dy}{dx} = \frac{x}{y}$.

  1. $y^2 = x^2 + C$
  2. $y = x^2 + C$
  3. $y = \ln(x) + C$
  4. $y = x + C$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Separate the variables and integrate both sides to obtain the general solution.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{y}{x} + \frac{x}{y}$. Which of the following is a suitable substitution to solve this equation?

  1. $y = vx$
  2. $y = x^v$
  3. $y = \ln(x)$
  4. $y = e^x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Substituting $y = vx$ transforms the equation into a separable equation.

Multiple choice

Find the general solution of the differential equation $\frac{dy}{dx} = \frac{2x + y}{x}$.

  1. $y = x^2 + C$
  2. $y = x + C$
  3. $y = \ln(x) + C$
  4. $y = \frac{1}{x} + C$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This equation is a linear first order differential equation. Solve it using the integrating factor method.

Multiple choice

Which of the following is a homogeneous first order differential equation?

  1. $y' + y = x$
  2. $y' = xy$
  3. $y'' + 2y' + y = 0$
  4. $y' = \frac{y}{x}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A homogeneous first order differential equation is one in which the dependent variable and its derivatives appear only in terms of the independent variable.

Multiple choice

Solve the following homogeneous differential equation: $\frac{dy}{dx} = \frac{y - x}{y + x}$.

  1. $y = x + C$
  2. $y = x^2 + C$
  3. $y = \ln(x) + C$
  4. $y = \frac{1}{x} + C$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This equation is a homogeneous first order differential equation. Solve it by substituting $y = vx$.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$. Which of the following is a suitable substitution to solve this equation?

  1. $y = vx$
  2. $y = x^v$
  3. $y = \ln(x)$
  4. $y = e^x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Substituting $y = vx$ transforms the equation into a separable equation.

Multiple choice

Find the general solution of the differential equation $\frac{dy}{dx} = \frac{2x + 3y}{x + 2y}$.

  1. $y = x + C$
  2. $y = x^2 + C$
  3. $y = \ln(x) + C$
  4. $y = \frac{1}{x} + C$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This equation is a linear first order differential equation. Solve it using the integrating factor method.