Mathematics · Physics

Differential Equations

247 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 two solutions x(t) = x1(t) and x(t) = x2 (t) of the differential equation $\frac{d^2x(t)}{dt^2}$ + x (t) = 0, t > 0, such that x1 (0) = 1,$\frac{dx_1(t)}{dt} \Big|{t -0} = 0, x_2(0) = 0, \frac{dx_2(t)}{dt}\Big|{t-0}$ = 1

The Wronskian W (t) = $\begin{bmatrix} \ x_1(t) & x_2(t) \ \ \frac{dx_1(t)}{dt} & \frac{dx_2(t)}{dt} \ \end{bmatrix}$ at t = $\frac{\pi}{2}$ is

  1. 1

  2. – 1

  3. 0
  4. 2$\pi$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Correct option is (1).

Multiple choice

The differential equation governing the vibrating system is
  1. m$\ddot X $ + c$\ddot X $ + k (x − y) = 0
  2. m ($\ddot X $ - $\ddot y $) + c ($\ddot X $ - $\dot y $) + kx = 0
  3. m $\ddot X $ + c ($\dot X $ - $\dot y $) + kx = 0
  4. m ($\ddot X $ - $\ddot y $) + c ($\dot X $ - $\dot y $) + k (x - y) = 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Assume any arbitrary relationship between the coordinates and their first derivatives, say $x>y$ and $\dot{x}>\dot{y}$, Alos assume $x>0$ and $\dot{x}>0$

Multiple choice business mathematics and statistics applications of calculus marginal income and marginal cost to find the maximum profit if marginal revenue and marginal cost function are given: integral calculus – ii

Given the marginal cost function $\dfrac{2x}{3}+3-\dfrac{16}{x^2}$, find  average cost function.

  1. $\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$
  2. $\dfrac{1}{2}x^2+3x-7+\dfrac{16}{x}$
  3. $\dfrac{1}{4}x^2+3x-7+\dfrac{16}{x}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\Rightarrow$  We have $MC=\dfrac{2x}{3}+3-\dfrac{16}{x^2}$


$\Rightarrow$  $Average\,\,cost=\int (MC)dx$ 


$\Rightarrow$   $Average\,\,cost=\int (\dfrac{2x}{3}+3-\dfrac{16}{x^2})dx$


$\therefore$   $Average\,\,cost=\dfrac{2x^2}{2\times 3}+3x-7+\dfrac{16}{x}$

$\Rightarrow$   $Average\,\,cost=\dfrac{1}{3}x^2+3x-7+\dfrac{16}{x}$

Multiple choice maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

The differential equation $\dfrac {dy}{dx}=\dfrac {1}{ax+by+c}$ where a,b,c are all non zero real number ,is

  1. Linear in $y$
  2. Linear in $x$
  3. linear in both $x$ and $y$

  4. Homogeneous equation
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac {dx}{dy}= ax+by+c$
$\dfrac {dx}{dy}-a=by+c$
Linear in $x$