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AP Intermediate 2nd Year Differential Equations — practice questions

Explore 88 free Differential Equations questions in AP Intermediate 2nd Year 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 88.

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 →A matrix with all elements zero is called:Read the answer and worked solution →Number of rows in matrix A times number of columns in A x B should equal:Read the answer and worked solution →Mode of {1, 2, 2, 3, 4, 4, 4, 5}:Read the answer and worked solution →Value of arctan(1):Read the answer and worked solution →Simplify arctan(sin x / (1+cos x)) for x ∈ (-π, π):Read the answer and worked solution →arccos(x) + arccos(-x) =Read the answer and worked solution →The ORDER and DEGREE of the DE (d²y/dx²)³ + (dy/dx)⁵ + y = 0 are:Read the answer and worked solution →The general solution of a 3rd-order DE contains how many arbitrary constants?Read the answer and worked solution →The DE dy/dx = y/x is best solved by:Read the answer and worked solution →For the linear DE dy/dx + (1/x)y = x, the INTEGRATING FACTOR μ(x) is:Read the answer and worked solution →If a population grows exponentially as dN/dt = 0.1 N (with t in years), the DOUBLING TIME is:Read the answer and worked solution →By Newton's law of cooling, a hot tea cup at 90°C in a room at 25°C cools so that: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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More Mathematics topics

Vector Algebra and 3D GeometryContinuity, Differentiability & DerivativesIntegrals and ApplicationsLinear ProgrammingRelations, Functions & Inverse TrigonometryDeterminantsMatrices and DeterminantsProbability