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 mathematics and statistics hyperbola introduction to hyperbola standard equation of hyperbola conic sections

General solution of the equation $ y=x\dfrac{dy}{dx}+\dfrac {dx}{dy}$ represents _____________.

  1. a straight line or hyperbola

  2. a straight line or parabola

  3. a parabola or hyperbola

  4. circles
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The given differential equation can be rearranged into a form that describes a family of curves, which are hyperbolas or parabolas depending on the integration constants.

Multiple choice physics parametric equations proving properties of curves derivatives - introduction and interpretation introduction to calculus - differentiation

The general solution of the differential equation $\sin{2x}\left( \cfrac { dy }{ dx } -\sqrt { \tan { x }  }  \right) -y=0$ is $y\phi(x)=x+c$ then ${ \Phi  }^{ 1 }\left( \cfrac { \pi  }{ 4 }  \right) $ is _____

  1. $1$
  2. $-1$
  3. $\cfrac{1}{\sqrt{3}}$
  4. $\sqrt{-3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\displaystyle \frac{dy}{dx}-\text{cosec}2x y=\sqrt{\tan x};I.F=e^{\displaystyle \int-\text{cosec}2xdx}$

$=\sqrt{\text{cosec }2x+\text{cot} 2x} \implies y\sqrt{\text{cosec}2x+\text{cot}2x}=x+c\implies \phi'\bigg(\frac{\pi}{4}\bigg)=-1$

Multiple choice physics parametric equations proving properties of curves derivatives - introduction and interpretation introduction to calculus - differentiation

The second order differential equation is :

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

We know that, the order of the differential equation is the order of highest derivative.
So, equation $y'+y''+y=\sin { x } $ is the second order differential equation.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

Which of the following is the solution of the equation $\displaystyle \frac{7y+4}{y+2}=\frac{-4}{3}$ ?

  1. $\displaystyle y = -\frac{4}{5}$
  2. $\displaystyle y = \frac{4}{5}$
  3. $\displaystyle y = -\frac{5}{4}$
  4. $\displaystyle y = \frac{5}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{7y+4}{y+2}=\dfrac{-4}{3}$
$21y+12=-4y-8$
$25y=-20$
$y=\dfrac{-20}{25} = \dfrac{-4}{5}$

Multiple choice

What is the general form of a first-order partial differential equation?

  1. F(x, y, z, p, q) = 0

  2. F(x, y, z, p, q, r) = 0

  3. F(x, y, z, p, q, r, s) = 0

  4. F(x, y, z, p, q, r, s, t) = 0
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The general form of a first-order partial differential equation is F(x, y, z, p, q) = 0, where p and q are the partial derivatives of the dependent variable z with respect to x and y, respectively.

Multiple choice

Which method is commonly used to solve linear partial differential equations with constant coefficients?

  1. Method of Characteristics

  2. Separation of Variables

  3. Laplace Transform

  4. Fourier Series
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Separation of Variables is a powerful technique for solving linear partial differential equations with constant coefficients. It involves finding solutions that are products of functions, each depending on a single independent variable.

Multiple choice

What is the fundamental solution of a linear partial differential equation?

  1. A solution that satisfies the equation at a single point

  2. A solution that satisfies the equation at all points in the domain

  3. A solution that is a linear combination of other solutions

  4. A solution that is obtained by applying a Laplace transform
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The fundamental solution of a linear partial differential equation is a solution that satisfies the equation at a single point. It is also known as the Green's function.

Multiple choice

What is the general form of a second-order linear partial differential equation?

  1. Au + Bv + Cw + Du + Ev + Fw + G = 0

  2. Au + Bv + Cw + Du + Ev + Fw = 0

  3. Au + Bv + Cw + Du + Ev = 0

  4. Au + Bv + Cw = 0
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The general form of a second-order linear partial differential equation is Au + Bv + Cw + Du + Ev + Fw + G = 0, where u, v, and w are the dependent variables, and A, B, C, D, E, F, and G are the coefficients.

Multiple choice

What is the Laplace transform used for in solving partial differential equations?

  1. To find the general solution of a partial differential equation

  2. To find the particular solution of a partial differential equation

  3. To find the fundamental solution of a partial differential equation

  4. To find the solution of a partial differential equation with boundary conditions
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Laplace transform is a powerful tool for solving partial differential equations with boundary conditions. It involves transforming the partial differential equation into an algebraic equation, which can then be solved using standard techniques.

Multiple choice

What is the general form of a first-order linear ODE?

  1. $y' + p(x)y = q(x)$
  2. $y'' + p(x)y' + q(x)y = 0$
  3. $y''' + p(x)y'' + q(x)y' + r(x)y = 0$
  4. $y^{(n)} + p_1(x)y^{(n-1)} + ... + p_n(x)y = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The general form of a first-order linear ODE is $y' + p(x)y = q(x)$, where $p(x)$ and $q(x)$ are functions of the independent variable $x$.

Multiple choice

Which method is commonly used to solve a second-order linear ODE with constant coefficients?

  1. Separation of Variables

  2. Method of Undetermined Coefficients

  3. Variation of Parameters

  4. Laplace Transform
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The Method of Undetermined Coefficients is a standard technique for solving second-order linear ODEs with constant coefficients.

Multiple choice

What is the general solution of the differential equation $\frac{dy}{dx} = 2x + 1$?

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

To solve this differential equation, we can use the method of integrating factors. The integrating factor is $e^{\int 2x dx} = e^{x^2}$. Multiplying both sides of the differential equation by the integrating factor, we get $e^{x^2} \frac{dy}{dx} = 2xe^{x^2} + e^{x^2}$. Now, we can integrate both sides to get $e^{x^2} y = 2e^{x^2} + C$, where $C$ is the constant of integration. Dividing both sides by $e^{x^2}$, we get $y = x^2 + x + C$, which is the general solution.

Multiple choice

What is the particular solution of the differential equation $\frac{dy}{dx} = 2x + 1$, given the initial condition $y(0) = 2$?

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

To find the particular solution, we can use the general solution $y = x^2 + x + C$ and the initial condition $y(0) = 2$. Substituting $x = 0$ and $y = 2$ into the general solution, we get $2 = 0^2 + 0 + C$, which implies that $C = 2$. Therefore, the particular solution is $y = x^2 + x + 2$.