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PUCET (Punjab) Differential Equations — practice questions

Explore 73 free Differential Equations questions in PUCET (Punjab) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Differential Equations questions and exercises with answers

Showing questions 1–30 of 73.

The **order** of a differential equation is:Read the answer and worked solution →The order and degree of $x^2 \dfrac{d^2y}{dx^2} + 3x \dfrac{dy}{dx} + 4y = 0$ are respectively:Read the answer and worked solution →The **general solution** of a differential equation contains:Read the answer and worked solution →The integrating factor for the linear DE $\dfrac{dy}{dx} + P(x)y = Q(x)$ is:Read the answer and worked solution →To solve a **homogeneous** differential equation of degree 1 (e.g. $dy/dx = f(y/x)$), the standard substitution is:Read the answer and worked solution →The order of the differential equation obtained by eliminating the arbitrary constants $A$ and $B$ from $y = A e^{3x} + B e^{-3x}$ is:Read the answer and worked solution →The order and degree of $\sqrt{1 + (dy/dx)^2} = d^2y/dx^2$ are:Read the answer and worked solution →Eliminate the arbitrary constant from $y^2 = 4ax$ to form the DE:Read the answer and worked solution →The DE obtained by eliminating $c_1$ and $c_2$ from $y = c_1 e^{3x} + c_2 e^{2x}$ is:Read the answer and worked solution →Solve $\dfrac{dy}{dx} = \dfrac{1 + y^2}{1 + x^2}$.Read the answer and worked solution →Solve $\dfrac{dy}{dx} = e^{x+y}$.Read the answer and worked solution →The DE $y - x \dfrac{dy}{dx} = 0$ has general solution:Read the answer and worked solution →The general solution of $\dfrac{dy}{dx} + y = e^{-x}$ is:Read the answer and worked solution →Solve the homogeneous DE $x\,dy/dx = x\,\tan(y/x) + y$:Read the answer and worked solution →The differential equation of the family of straight lines $y = mx + c$ (with $m, c$ arbitrary) is:Read the answer and worked solution →Solve $\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0$:Read the answer and worked solution →Solve $\dfrac{dy}{dx} = \dfrac{y + \sqrt{x^2 + y^2}}{x}$.Read the answer and worked solution →The half-life of a radioactive substance decaying by $dm/dt = -km$ (with $k > 0$) in terms of $k$ is:Read the answer and worked solution →Newton's law of cooling: $dT/dt = -k(T - T_s)$ where $T_s$ is the surrounding temperature. The general solution is:Read the answer and worked solution →Solve $(x^2 - y^2)\,dx + 2xy\,dy = 0$.Read the answer and worked solution →The particular solution of $\dfrac{dy}{dx} + y\tan x = \sec x$ given $y(0) = 1$ is:Read the answer and worked solution →If $y = e^x \cos x$, then $\dfrac{d^2 y}{dx^2} - 2\dfrac{dy}{dx} + 2y$ equals:Read the answer and worked solution →Order of differential equation:Read the answer and worked solution →General solution of dy/dx = k y:Read the answer and worked solution →Variable separable form:Read the answer and worked solution →For dy/dx = x, general solution:Read the answer and worked solution →Solve dy/dx = (y/x):Read the answer and worked solution →Linear DE of first order: dy/dx + P(x) y = Q(x). Solution uses:Read the answer and worked solution →Integrating factor for dy/dx + y = e^x:Read the answer and worked solution →Solve dy/dx = -y/x with y(1) = 2:Read the answer and worked solution →

Worked practice questions

The **order** of a differential equation is:

  1. The highest power of the dependent variable
  2. The highest-order derivative present in the equation
  3. The number of arbitrary constants in the solution
  4. Always equal to its degree
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Correct answer: B. The highest-order derivative present in the equation

Order = highest order of derivative appearing. e.g. $d^2y/dx^2 + 3(dy/dx) + y = 0$ has order 2 (because of $d^2y/dx^2$). The order is always a positive integer.

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The order and degree of $x^2 \dfrac{d^2y}{dx^2} + 3x \dfrac{dy}{dx} + 4y = 0$ are respectively:

  1. 1, 1
  2. 2, 1
  3. 2, 2
  4. 1, 2
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Correct answer: B. 2, 1

Highest-order derivative is $d^2y/dx^2$ → order 2. Its power is 1 → degree 1.

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The **general solution** of a differential equation contains:

  1. No arbitrary constants
  2. As many arbitrary constants as the order of the equation
  3. Exactly one arbitrary constant always
  4. More constants than the order of the equation
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Correct answer: B. As many arbitrary constants as the order of the equation

An $n$-th order DE has a general solution with $n$ arbitrary constants. Particular solutions arise when constants are fixed by initial/boundary conditions.

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