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Chapter 9 · Differential Equations

Exercise 9.2 · Q11 (MCQ): Arbitrary constants in the general solution of a fourth order equation

Exercise 9.2General and particular solutions · Section 9.3
The question

11. The number of arbitrary constants in the general solution of a differential equation of fourth order are:

(A) 0    (B) 2    (C) 3    (D) 4

Exam answer

The general solution of a differential equation contains as many arbitrary constants as the order of the equation.

The order is 4, so the number of arbitrary constants is 4.

Answer: (D) 4

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

The one idea this problem is built on

Solving an equation of order nn needs, in effect, nn integrations to get from the nnth derivative back to yy⁠, and each integration brings one arbitrary constant. So the general solution of a fourth order equation carries exactly 4 arbitrary constants, option (D).

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

Basics you need

1General and particular solutions

A general solution contains arbitrary constants; a particular solution is free from them (Section 9.3). The Summary of Chapter 9 adds: a general solution has as many arbitrary constants as the order. Lesson: general and particular solutions

2Order

The order of a differential equation is the order of its highest order derivative (Section 9.2.1). Lesson: order and degree

3Each integration adds a constant

∫0 dx=C\int 0\,dx = C⁠, ∫C dx=Cx+D\int C\,dx = Cx + D⁠, and so on. Basics: order n gives n constants

Solution

  1. What a general solution is

    Section 9.3 calls a solution with arbitrary constants a general solution, and the Summary of the chapter makes the count precise: the general solution contains as many arbitrary constants as the order of the equation.

    The question asks for a number, so we need the rule that fixes the number, not only the definition.
    General solution, Section 9.3 and the Summary. Lesson: general and particular solutions

  2. See it on the simplest fourth order equation

    Take d4ydx4=0\frac{d^4y}{dx^4} = 0⁠. Integrating four times, one constant each time:

    d3ydx3=C1d2ydx2=C1x+C2dydx=12C1x2+C2x+C3y=16C1x3+12C2x2+C3x+C4.\begin{aligned} \frac{d^3y}{dx^3} &= C_1 \\ \frac{d^2y}{dx^2} &= C_1x + C_2 \\ \frac{dy}{dx} &= \tfrac{1}{2}C_1x^2 + C_2x + C_3 \\ y &= \tfrac{1}{6}C_1x^3 + \tfrac{1}{2}C_2x^2 + C_3x + C_4. \end{aligned}

    Four independent constants C1,C2,C3,C4C_1, C_2, C_3, C_4⁠: one per integration.

    Going from the fourth derivative back to yy takes four integrations, and each one adds a constant that differentiation would destroy. None of the four can be merged into another, because they multiply different powers of xx⁠.
    Each integration adds one constant. Basics: order n gives n constants

  3. Choose the option

    Order 4, so the general solution has 4 arbitrary constants. The correct option is (D).

    The rule of Step 1 applies to any fourth order equation, and Step 2 shows where the four come from.
    The twelve multiple-choice questions of the chapter. Lesson: the multiple-choice questions

  4. Examine every option

    OptionWhere the number comes fromVerdict
    (A) 0The count for a particular solutionWrong: a general solution has constants
    (B) 2The count for a second order equationWrong: the order here is 4
    (C) 3One integration too fewWrong: four integrations give four constants
    (D) 4As many constants as the orderCorrect
    (A) mixes up general and particular solutions; Q12 asks exactly that other question. (B) would fit an equation like y′′+y=0y'' + y = 0⁠, whose general solution acos⁡x+bsin⁡xa\cos x + b\sin x has two constants. (C) loses one integration. Only (D) follows the rule.
    General versus particular solutions (Section 9.3). Lesson: Examples 2 and 3
Result

A general solution has as many arbitrary constants as the order, so a fourth order equation has 4: option (D).

(Textbook answer: D.)

Check it with numbers

Choose C1=6C_1 = 6⁠, C2=4C_2 = 4⁠, C3=3C_3 = 3⁠, C4=4C_4 = 4 in Step 2: y=x3+2x2+3x+4y = x^3 + 2x^2 + 3x + 4⁠. Its values at x=0,1,2,3,4x = 0, 1, 2, 3, 4 are 4,10,26,58,1124, 10, 26, 58, 112⁠. The steps are 6,16,32,546, 16, 32, 54⁠, then 10,16,2210, 16, 22⁠, then 6,66, 6⁠, then 00⁠. The fourth steps vanish, the counting version of d4ydx4=0\frac{d^4y}{dx^4} = 0⁠, whatever values the four constants take.

Common mistakes
  1. Mixing up general and particular. General: as many constants as the order; particular: none.
  2. Linking the count to the degree. The number of constants follows the order, not the degree.
What to take away
  1. Order nn means nn arbitrary constants in the general solution.
  2. A particular solution has none: see Q12.

Every fact used, and where it comes from

FactUsed inSource
General solution: arbitrary constants; particular: noneSteps 1, 4Section 9.3 (Lesson)
As many arbitrary constants as the orderSteps 1, 3Summary, Chapter 9 (Basics)
Order = order of the highest derivativeStep 2Section 9.2.1
Each integration adds one constantStep 2Chapter 7 (integrals)
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