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

The solution of the ordinary differential equation $\frac{dy}{dx} + 2y$= 0 for the boundary condition, y = 5 at x = 1 is

  1. y = e - 2x

  2. y = 2e - 2x

  3. y = 10.95 e - 2x

  4. y = 36.95 e - 2x
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The differential equation dy/dx + 2y = 0 has the general solution y = C * e^(-2x). Using the boundary condition y(1) = 5, we get 5 = C * e^(-2), so C = 5 * e^2. Thus, y = 5 * e^2 * e^(-2x) = 5 * e^(2 - 2x). 5 * e^2 is approximately 36.95.

Multiple choice

The solution of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x$, with the condition that y = 1 at x = 1, is
  1. $y = \frac{2}{3x^2} + \frac{x}{3}$
  2. $y = \frac{x}{2} + \frac{1}{2x}$
  3. $y = \frac{2}{3} + \frac{x}{3}$
  4. $y = \frac{2}{3x} + \frac{x^2}{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

This is a linear differential equation dy/dx + (1/x)y = x. The integrating factor is e^(integral 1/x dx) = x. Multiplying by x gives d/dx(xy) = x^2. Integrating both sides gives xy = x^3/3 + C. Using y(1) = 1, we get 1 = 1/3 + C, so C = 2/3. Thus, xy = x^3/3 + 2/3, or y = x^2/3 + 2/(3x).

Multiple choice

The order and degree of the differential equation $\frac{d^3y}{dx^3} + 4 \sqrt{(\frac{dy}{dx})^3 + y^3 = 0}$ are respectively

  1. 3 and 2

  2. 2 and 3

  3. 3 and 3

  4. 3 and 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The order of a differential equation is the highest derivative present, which is 3 here. To find the degree, we must make the equation polynomial in derivatives; squaring both sides of the radical term (d^3y/dx^3 + 4 * sqrt(...) = 0) leads to a term involving (d^3y/dx^3)^2, making the degree 2.

Multiple choice

The solution for the differential equation $\frac{dy}{dx} = x^2y$ with condition that y = 1 at x = 0 is
  1. $y = e^{\frac{1}{2x}}$
  2. In $(y) = \frac{x^3}{3} + 4$
  3. In $(y) = \frac{x^2}{2}$
  4. $y = e^{\frac{x^3}{3}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Separating variables gives dy/y = x^2 dx. Integrating both sides results in ln(y) = x^3/3 + C. Using the condition y=1 at x=0, we find ln(1) = 0 + C, so C = 0. Thus, ln(y) = x^3/3, which implies y = e^(x^3/3).

Multiple choice

The general solution of $\frac{d^2y}{dx^2} + y = o $ is

  1. y = P cos x + Q sin x

  2. y = P cos x

  3. y = P sin x

  4. y = P sin2x
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The differential equation y'' + y = 0 has the characteristic equation r^2 + 1 = 0, which gives roots r = +/- i. The general solution is y = c1 cos(x) + c2 sin(x).

Multiple choice

The equation $k_x\frac{\partial^2h}{\partial x^2} + k_z \frac{\partial ^2h}{\partial z^2} = 0$can be transformed to$\frac{\partial^2h}{\partial x_1^2} + \frac{\partial ^2h}{\partial z^2} = 0$by substituting
  1. $x_1 = x \frac{k_z}{k_x}$
  2. $x_1 = x \frac{k_x}{k_z}$
  3. $x_1 = x \sqrt{\frac{k_x}{k_z}}$
  4. $x_1 = x \sqrt{\frac{k_z}{k_x}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To transform the equation kx(d^2h/dx^2) + kz(d^2h/dz^2) = 0 into the Laplace equation, we need to scale the x-coordinate. Substituting x1 = x * sqrt(kz/kx) results in the desired form.