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Chapter 9 · Differential Equations

Exercise 9.2 · Q12 (MCQ): Arbitrary constants in a particular solution of a third order equation

Exercise 9.2General and particular solutions · Section 9.3
The question

12. The number of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3    (B) 2    (C) 1    (D) 0

Exam answer

A particular solution is obtained from the general solution by giving particular values to all the arbitrary constants, so it is free from arbitrary constants.

Hence the number is 0, whatever the order.

Answer: (D) 0

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

The word "particular" is the whole answer. A particular solution is what you get after every arbitrary constant of the general solution has been given a definite value, so none is left. The order, three, decides how many constants the general solution has (3), not the particular one (0). Option (D).

Given, to find, and the plan
Given
A differential equation of order 3.
To find
The number of arbitrary constants in a particular solution, and which option gives it.
Plan
(1) Recall the definition of a particular solution. (2) See it on a simple third order equation. (3) Choose the option. (4) Examine every option.

Basics you need

1Particular solution

"The solution free from arbitrary constants", obtained from the general solution by giving particular values to the arbitrary constants (Section 9.3). Lesson: general and particular solutions

2General solution

Contains as many arbitrary constants as the order (Summary, Chapter 9). Basics: order n gives n constants

Solution

  1. The definition

    Section 9.3: the solution free from arbitrary constants, that is, the solution obtained from the general solution by giving particular values to the arbitrary constants, is called a particular solution.

    "Free from arbitrary constants" means the count is zero. The definition does not depend on the order.
    Section 9.3, general and particular solutions. Lesson: general and particular solutions

  2. See it on a third order equation

    For d3ydx3=0\frac{d^3y}{dx^3} = 0⁠, integrating three times (and writing C1C_1 for half of the first constant) gives the general solution

    y=C1x2+C2x+C3,y = C_1x^2 + C_2x + C_3,

    with three arbitrary constants, as many as the order. Giving them values, for example C1=1C_1 = 1⁠, C2=0C_2 = 0⁠, C3=0C_3 = 0⁠, gives the particular solution y=x2y = x^2⁠, with no arbitrary constant left.

    The general solution has 3 constants; the particular solution is one member of that family, chosen by fixing all of them, so it has 0.
    Each integration adds one constant. Basics: order n gives n constants

  3. Choose the option

    A particular solution has 0 arbitrary constants. The correct option is (D).

    The order 3 is a distractor for this question; it matters only for the general solution.
    The twelve multiple-choice questions of the chapter. Lesson: the multiple-choice questions

  4. Examine every option

    OptionWhere the number comes fromVerdict
    (A) 3The count for the general solutionWrong: a particular solution has none
    (B) 2Fixing only one of the three constantsWrong: that is still not a particular solution
    (C) 1Fixing two of the three constantsWrong: all must be fixed
    (D) 0Free from arbitrary constantsCorrect
    (A) answers Q11's kind of question (general solution) for order 3. (B) and (C) describe partly fixed families, such as y=x2+C2x+C3y = x^2 + C_2x + C_3⁠, which still contain arbitrary constants. Only (D) matches the definition.
    General versus particular solutions (Section 9.3). Lesson: Examples 2 and 3
Result

A particular solution is free from arbitrary constants, so it has 0 of them, whatever the order: option (D).

(Textbook answer: D.)

Check it with numbers

The particular solution y=x2y = x^2 of d3ydx3=0\frac{d^3y}{dx^3} = 0 takes the values 0,1,4,90, 1, 4, 9 at x=0,1,2,3x = 0, 1, 2, 3⁠. The steps are 1,3,51, 3, 5⁠, then 2,22, 2⁠, then 00⁠: the third steps vanish, as d3ydx3=0\frac{d^3y}{dx^3} = 0 requires. Every value is a fixed number; nothing in it can still be chosen.

Common mistakes
  1. Answering with the order (A). That counts the constants of the general solution.
  2. Thinking a particular solution keeps some constants. All of them are given values.
What to take away
  1. General solution: as many constants as the order. Particular solution: none.
  2. Pair this with Q11.

Every fact used, and where it comes from

FactUsed inSource
Particular solution: free from arbitrary constantsSteps 1, 3, 4Section 9.3 (Lesson)
General solution has as many constants as the orderSteps 2, 4Summary, Chapter 9 (Basics)
Each integration adds one constantStep 2Chapter 7 (integrals)
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