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AP Intermediate 2nd Year Differential Equations — practice questions — page 3

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 61–88 of 88.

Match each method to the form it solvesRead the answer and worked solution →A mass on a spring obeys d²x/dt² + ω²x = 0 (SHM). The period of oscillation is: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 →Spot the odd one outRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Reading definitions backRead the answer and worked solution →Tap every statement true of a general solutionRead the answer and worked solution →Which belongs to which idea?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 →True or false: order and degreeRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →The DE dy/dx = y/x is best solved by:Read the answer and worked solution →Order the steps: solve by separating variablesRead 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 →Order the reasoning: from equation to particular solutionRead the answer and worked solution →True or false: the Historical NoteRead the answer and worked solution →Counting arbitrary constantsRead the answer and worked solution →Match each term to its meaningRead the answer and worked solution →Flip and pair: form with nameRead the answer and worked solution →True or false: what a solution isRead the answer and worked solution →General solution or particular solution?Read the answer and worked solution →Recognising the typeRead the answer and worked solution →True or false: methods of solutionRead the answer and worked solution →Order, degree, or neither?Read the answer and worked solution →Complete the definition of a differential equationRead the answer and worked solution →Complete the definition of order and degreeRead 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 →

Worked practice questions

Match each method to the form it solves

Read exercise and reference answers
Matching pairs
  1. Left
    Variable separable
    Right
    all y terms go with dy, all x terms with dx
  2. Left
    Homogeneous method
    Right
    dy/dx = f(x, y) with f homogeneous of degree zero
  3. Left
    Linear method
    Right
    dy/dx + Py = Q with P, Q functions of x only
Three methods, three recognisable forms. Choosing correctly is most of the chapter. Source: syllabus Maths Part II, Ch 9 Differential Equations.

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A mass on a spring obeys d²x/dt² + ω²x = 0 (SHM). The period of oscillation is:

  1. $2\pi \omega$ (inverse)
  2. $T = 2\pi/\omega$ (the period formula)
  3. $T = 1/\omega$ (frequency reciprocal)
  4. $T = \omega^2$ (squared)
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Correct answer: B. $T = 2\pi/\omega$ (the period formula)

**SHM:** x(t) = A cos(ωt + φ). Period T = 2π/ω. Frequency f = ω/(2π) = 1/T.

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The general solution of a 3rd-order DE contains how many arbitrary constants?

  1. One arbitrary constant
  2. Two arbitrary constants
  3. Three arbitrary constants
  4. Six arbitrary constants
Show answer and solution

Correct answer: C. Three arbitrary constants

Number of arbitrary constants in general solution = order of the DE. For 3rd-order, 3 constants. Particular solution fixes them via 3 initial conditions.

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