Class 12 Maths, Worked in Full Past papersGet updates
Chapter 9 · Differential Equations

Exercise 9.1 · Q6: Order and degree of (y′′′)2+(y′′)3+(y′)4+y5=0(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0

Exercise 9.1Order and degree · Sections 9.2.1 and 9.2.2
The question

Determine order and degree (if defined) of differential equations given in Exercises 1 to 10.

6. (y′′′)2+(y′′)3+(y′)4+y5=0(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0

Exam answer

The highest order derivative present in the equation is y′′′y'''⁠, so its order is 3.

The equation is a polynomial equation in its derivatives y′′′y'''⁠, y′′y'' and y′y'⁠.

The highest power of y′′′y''' is 2, so its degree is 2.

∴\therefore Order 3; degree 2.

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

The one idea this problem is built on

Four terms carry four powers: 2, 3, 4 and 5. Only one of them matters for the degree, the power on the highest derivative. The highest derivative is y′′′y'''⁠, so the order is 3. The equation is a polynomial in y′′′y'''⁠, y′′y'' and y′y'⁠, so the degree is defined, and it is the power on y′′′y'''⁠, which is 2.

Given, to find, and the plan
Given
The differential equation (y′′′)2+(y′′)3+(y′)4+y5=0(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0⁠, with yy the dependent variable and xx the independent variable.
To find
Its order, and its degree if the degree is defined.
Plan
(1) List every derivative in the equation. (2) The order is the highest of them. (3) Test whether the equation is a polynomial equation in its derivatives. (4) Read off the power of the highest derivative only.

Basics you need

1Dashes and powers

The dashes count differentiations: y′y'⁠, y′′y''⁠, y′′′y''' are the first, second and third derivatives (Note 1, Section 9.2). A power is written outside a bracket: (y′′)3(y'')^3 is the second derivative cubed. Basics: y′ and y″

2Order

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

3Degree, and its condition

The degree is defined only when the equation is a polynomial equation in its derivatives. Then it is the highest power (a positive integer) of the highest order derivative (Section 9.2.2). Lesson: order and degree

Solution

  1. List the derivatives and their powers

    Term by term:

    (y′′′)2(y''')^2⁠: the third derivative, to the power 2.

    (y′′)3(y'')^3⁠: the second derivative, to the power 3.

    (y′)4(y')^4⁠: the first derivative, to the power 4.

    y5y^5⁠: yy itself, to the power 5; this is not a derivative.

    Separating "which derivative" (the dashes) from "what power" (the number outside the bracket) is the whole question. Writing them term by term keeps the two apart.
    Notation y′y'⁠, y′′y''⁠, y′′′y'''⁠: Note 1, Section 9.2. Basics: y′ and y″

  2. Read off the order

    The highest order derivative present is y′′′y'''⁠. So

    order=3.\text{order} = 3.
    Section 9.2.1: the order is the order of the highest order derivative. Three dashes is the most, so the order is 3.
    Definition of order, Section 9.2.1. Lesson: order and degree

  3. Is it a polynomial equation in its derivatives?

    Write p=y′p = y'⁠, q=y′′q = y''⁠, r=y′′′r = y'''⁠. The equation reads

    r2+q3+p4+y5=0.r^2 + q^3 + p^4 + y^5 = 0.

    Every derivative appears with a whole-number power, and none sits inside sin, cos, tan, log or e. So this is a polynomial equation in its derivatives, and its degree is defined.

    The degree is only defined for a polynomial equation in the derivatives (Section 9.2.2). The term y5y^5 is a polynomial in yy too, but even a non-polynomial term in yy alone would not matter, because the condition is about the derivatives.
    Condition for the degree, Section 9.2.2. Lesson: Example 1, order and degree

  4. Read off the degree

    The highest order derivative is r=y′′′r = y'''⁠, and its highest power in the equation is 2. So

    degree=2.\text{degree} = 2.
    By Section 9.2.2, the degree is the highest power of the highest order derivative. The powers 3 and 4 belong to the lower derivatives y′′y'' and y′y'⁠, and the 5 belongs to yy⁠, which is not a derivative at all; none of them counts.
    Definition of degree, Section 9.2.2. Lesson: Exercise 9.1 player
Result

The equation (y′′′)2+(y′′)3+(y′)4+y5=0(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0 has order 3 and degree 2. It is a polynomial equation in its derivatives, so the degree is defined.

(Textbook answer: Order 3; Degree 2.)

Check it with numbers

Hold y′′=0.5y'' = 0.5⁠, y′=1y' = 1 and y=−1y = -1 fixed. Then (0.5)3+14+(−1)5(0.5)^3 + 1^4 + (-1)^5 is 0.125+1−1=0.1250.125 + 1 - 1 = 0.125⁠, and the left side is (y′′′)2+0.125(y''')^2 + 0.125⁠.

At y′′′=0,1,2,3y''' = 0, 1, 2, 3 it is 0.1250.125⁠, 1.1251.125⁠, 4.1254.125⁠, 9.1259.125⁠. The steps are 1,3,51, 3, 5⁠, and the steps of those are 2,22, 2⁠. Constant second steps are the mark of a square, so y′′′y''' enters to the power 2: degree 2.

Common mistakes
  1. Answering "degree 5" (or 4). The largest power on the page belongs to yy (or to y′y'⁠), not to the highest derivative.
  2. Answering "order 2" from the square. The square is a power; the order comes from the three dashes.
  3. Counting yy as a derivative. y5y^5 has no dash; it affects neither the order nor the degree.
What to take away
  1. Order: count the dashes on the most-dashed derivative. Degree: the power on that same derivative.
  2. Powers of lower derivatives and of yy never decide the degree.

Every fact used, and where it comes from

FactUsed inSource
y′y'⁠, y′′y''⁠, y′′′y''' are the first, second, third derivativesStep 1Note 1, Section 9.2
Order = order of the highest order derivativeStep 2Section 9.2.1 (Lesson: order and degree)
Degree is defined only for a polynomial equation in the derivativesStep 3Section 9.2.2
Degree = highest power of the highest order derivativeStep 4Section 9.2.2
Found an error, or want more?

Tell us if something on this page is wrong or unclear, or if you would like a fuller explanation, another method or a new feature. Send feedback on this problem →