9. Differential Equations

7 topics2 h

Chapter 9 of NCERT Mathematics for Class 12: Differential Equations. Three sections — Equations whose unknown is a function, What counts as a solution and Three methods for first order, first degree equations. Seven videos, 2 h in all.

Worked answers to the 98 exercise questions in this chapter: Exercise 9.1 · Exercise 9.2 · Exercise 9.3 · Exercise 9.4 · Exercise 9.5 · Miscellaneous Exercise

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Part 1

Equations whose unknown is a function

16 minWhat makes an equation differential, and why the answer is a curve not a number16 min · 14 parts15 minOrder and degree, and the polynomial condition degree needs before it means anything15 min · 15 parts

Part 2

What counts as a solution

16 minGeneral against particular: why the arbitrary constants are counted by the order16 min · 16 parts

Part 3

Three methods for first order, first degree equations

19 minPushing all the y's one side and all the x's the other, then integrating19 min · 18 parts16 minHomogeneous equations, and the substitution that separates the variables for you16 min · 19 parts20 minMultiplying through by an integrating factor to make the left side a single derivative20 min · 21 parts20 minThe mirror-image form, where x is treated as the dependent variable20 min · 21 parts

What you will be able to do

  • Decide, for a given equation, whether a derivative of the unknown appears in it, and classify the equation accordingly
  • Find the highest derivative in a given equation and state the order
  • Explain why the answer to a differential equation is a function and not a number, and say what substituting it is supposed to achieve
  • Recognise, from the shape of the right-hand side, whether an equation can be separated
  • Test a two-variable expression for homogeneity by scaling both variables
  • Recognise the standard shape and name its two coefficient positions
  • Recognise when an equation is easier to read with x as the unknown

What this chapter assumes you already know

  • Differentiation of standard functions, from Class XI and from Chapter 5 of Part II's companion volume
  • Indefinite integration as the reverse of differentiation, from Chapter 7
  • The meaning of dependent and independent variable in a function
  • Function notation, and evaluating a function at a point
  • Sketching a curve from its equation, and reading a point off it
  • Solving an ordinary algebraic or trigonometric equation for a number
  • Higher derivatives, and what taking a derivative twice or three times means
  • The three notations for a derivative fixed in the previous topic

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A differential equation is any equation with a d in it."
  • "I am solving for x."
  • "So the answer is a number after all, once I finish integrating."
  • "Ordinary means easy and partial means hard."
  • "The prime notation and the fraction notation mean different things."
  • "A subscript on y means a term of a sequence."
  • "Verifying a solution and finding one are the same skill."
  • "If the two sides agree at one value of x, the function is a solution."