Mathematics · 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

In a study of the spread of a contagious disease, the number of infected individuals is given by the differential equation $\frac{dI}{dt} = \beta I (1 - \frac{I}{N})$, where $\beta$ is the transmission rate, $I$ is the number of infected individuals, and $N$ is the total population. What is the general solution to this differential equation?

  1. $I(t) = N(1 - e^{-\beta t})$
  2. $I(t) = \frac{N}{1 + e^{-\beta t}}$
  3. $I(t) = \frac{N}{1 - e^{-\beta t}}$
  4. $I(t) = N e^{-\beta t}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The general solution to the differential equation $\frac{dI}{dt} = \beta I (1 - \frac{I}{N})$ is $I(t) = \frac{N}{1 + e^{-\beta t}}$. This can be obtained by using the method of separation of variables.

Multiple choice

Which of the following differential equations models exponential population growth?

  1. $$\frac{dN}{dt} = rN$$
  2. $$\frac{dN}{dt} = rN(1 - \frac{N}{K})$$
  3. $$\frac{dN}{dt} = rN^2$$
  4. $$\frac{dN}{dt} = rN(N - K)$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Exponential population growth is characterized by a constant rate of growth proportional to the population size.

Multiple choice

What is the general solution to the differential equation $$\frac{dN}{dt} = rN$$?

  1. $$N(t) = N_0e^{rt}$$
  2. $$N(t) = N_0e^{-rt}$$
  3. $$N(t) = N_0 + rt$$
  4. $$N(t) = N_0 - rt$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The general solution to this differential equation is an exponential function.

Multiple choice

Which of the following differential equations models the spread of an infectious disease?

  1. $$\frac{dS}{dt} = -\beta SI$$
  2. $$\frac{dS}{dt} = \beta SI$$
  3. $$\frac{dS}{dt} = -\beta S^2$$
  4. $$\frac{dS}{dt} = \beta S^2$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The spread of an infectious disease is often modeled using a differential equation that includes a term representing the rate of infection and a term representing the rate of recovery.

Multiple choice

Which of the following differential equations models predator-prey interactions?

  1. $$\frac{dN_1}{dt} = r_1N_1 - \alpha_1N_1N_2$$
  2. $$\frac{dN_1}{dt} = r_1N_1 + \alpha_1N_1N_2$$
  3. $$\frac{dN_1}{dt} = -r_1N_1 + \alpha_1N_1N_2$$
  4. $$\frac{dN_1}{dt} = -r_1N_1 - \alpha_1N_1N_2$$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Predator-prey interactions are often modeled using a system of differential equations that includes terms representing the growth rate of the prey, the predation rate, and the carrying capacity of the environment.

Multiple choice

What is the general solution of the differential equation dy/dx = (x + y)/(x - y)?

  1. y = x + C

  2. y = x - C

  3. y = -x + C

  4. y = -x - C
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We can solve this differential equation using the method of separation of variables. Rewriting the equation as (x - y)dy = (x + y)dx, we can integrate both sides to get ∫(x - y)dy = ∫(x + y)dx. This gives us (1/2)x^2 - xy + C1 = (1/2)x^2 + xy + C2, where C1 and C2 are constants. Simplifying this equation, we get 2xy = C, where C = C2 - C1. Therefore, the general solution of the differential equation is y = x + C.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = y^2 + 1$. 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
B Correct answer
Explanation

The equation is Lipschitz continuous in $y$, so it has a unique solution for any initial condition in a sufficiently small interval. However, the equation is not globally Lipschitz continuous, so it may not have a unique solution for all initial conditions.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = y^{1/3}$. 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 $\frac{dy}{dx} = y^2 - 1$. 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
B Correct answer
Explanation

The equation is Lipschitz continuous in $y$, so it has a unique solution for any initial condition in a sufficiently small interval. However, the equation is not globally Lipschitz continuous, so it may not have a unique solution for all initial conditions.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{1}{y}$. 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
B Correct answer
Explanation

The equation is Lipschitz continuous in $y$, so it has a unique solution for any initial condition in a sufficiently small interval. However, the equation is not globally Lipschitz continuous, so it may not have a unique solution for all initial conditions.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = y - x$. 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
A Correct answer
Explanation

The equation is Lipschitz continuous in both $y$ and $x$, so it has a unique solution for any initial condition.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{y}{x}$. 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
A Correct answer
Explanation

The equation is Lipschitz continuous in both $y$ and $x$, so it has a unique solution for any initial condition.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = y^2 + x^2$. 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
B Correct answer
Explanation

The equation is Lipschitz continuous in $y$, so it has a unique solution for any initial condition in a sufficiently small interval. However, the equation is not globally Lipschitz continuous, so it may not have a unique solution for all initial conditions.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{x}{y}$. 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
B Correct answer
Explanation

The equation is Lipschitz continuous in $y$, so it has a unique solution for any initial condition in a sufficiently small interval. However, the equation is not globally Lipschitz continuous, so it may not have a unique solution for all initial conditions.

Multiple choice

Consider the differential equation $\frac{dy}{dx} = \frac{y}{x^2}$. 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
B Correct answer
Explanation

The equation is Lipschitz continuous in $y$, so it has a unique solution for any initial condition in a sufficiently small interval. However, the equation is not globally Lipschitz continuous, so it may not have a unique solution for all initial conditions.