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

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 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 $x$

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

The solution of the differential equation, $y\,dx + \left( {x + {x^2}y} \right)dy = 0$ is

  1. $\log y = cx$
  2. $ \log y- \dfrac{1}{{xy}} = c$
  3. $\dfrac{1}{{xy}} - \log y = c$
  4. $\dfrac{1}{{xy}} + \log y = c$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have,

$ydx+\left( { x+{ x^{ 2 } }y } \right) dy=0 \ ydx=-\left( { x+{ x^{ 2 } }y } \right) dy \ ydx+xdy=-{ x^{ 2 } }ydy \ \dfrac { { ydx+xdy } }{ { { { \left( { xy } \right)  }^{ 2 } } } } =-\dfrac { { dy } }{ y }  \ \dfrac { { d\left( { xy } \right)  } }{ { { { \left( { xy } \right)  }^{ 2 } } } } =-\dfrac { { dy } }{ y } $

On taking integrating both sides

$\begin{array}{l} \dfrac { { -1 } }{ { xy } } =-\log  y+c \\ \log  y-\dfrac { 1 }{ { xy } } =c \end{array}$


Hence, this is the answer.

Multiple choice physics option b: engineering physics introduction to sound free, forced and damped oscillations resonance

If we wish to represent the equation for the position of the mass in terms of a differential equation, which one of these would be the most suitable?

  1. $ m \dfrac{d^2x}{dt^2} + b \dfrac{dx}{dt} + kx = 0$
  2. $ m \dfrac{d^2x}{dt^2} - b \dfrac{dx}{dt} + kx = 0$
  3. $ m \dfrac{d^2x}{dt^2} + b \dfrac{dx}{dt} - kx = 0$
  4. $ m \dfrac{d^2x}{dt^2} -b \dfrac{dx}{dt} - kx = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The force on body oscillating in resistive medium is 

$f = -kx - bv$
$\Rightarrow m\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } } =-kx-b\dfrac { dx }{ dt } \ \Rightarrow m\dfrac { { d }^{ 2 }x }{ d{ t }^{ 2 } } +b\dfrac { dx }{ dt } +kx=0$
k = oscillating constant 
x = displacement of body from mean position 
b = constant depends on resistive medium 
v = velocity of object = $\dfrac{dx}{dt}$
m = mass of object .