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Forming a Differential Equation from a Family of Curves

Forming a differential equation means eliminating arbitrary constants from a given family of curves by differentiating enough times and combining the resulting equations.

In one line: Constants disappear by differentiation plus elimination; the final equation keeps x, y, and derivatives only.

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Understand it

A family of curves may contain arbitrary constants such as a, b, or c. A differential equation represents the whole family without explicitly containing those constants. The number of differentiations usually equals the number of arbitrary constants to be eliminated.

Example

For y = mx, where m is an arbitrary constant, differentiate: dy/dx = m. Since m = y/x from the original equation, dy/dx = y/x. Thus x(dy/dx) - y = 0 is the required differential equation.

Formula and rule

Method condition: if a family contains n independent arbitrary constants, differentiate n times and eliminate those constants.

The resulting differential equation generally has order n.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Example: Form the differential equation of y = ax + b.

  1. 1Differentiate once: dy/dx = a.
  2. 2Differentiate again: d2y/dx2 = 0.
  3. 3Since both arbitrary constants are removed, the required differential equation is d2y/dx2 = 0.

Method you can reuse

  1. 1List the arbitrary constants.
  2. 2Differentiate once for each constant.
  3. 3Express constants from the original and derivative equations, or combine equations to cancel them.
  4. 4Write the final relation involving x, y, and derivatives only.

Visual tip

A diagram is not required unless the exam question gives a geometric family of curves. For algebraic families, equations and derivatives are sufficient.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

How many times should y = ax + b normally be differentiated to form its differential equation?

Think first, then show answer
Twice, because the family contains two arbitrary constants, a and b.

Common wrong answer

Stopping after dy/dx = a for y = ax + b, which still contains the arbitrary constant a and therefore is not the required differential equation.

Don't mix this up

A common error is differentiating but leaving the arbitrary constant in the final answer. A formed differential equation should not contain the eliminated arbitrary constants.

Exam tip

Count the arbitrary constants first. Differentiate that many times, then use the original equation and derivative equations together to remove every arbitrary constant.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked as a short-answer or long-answer item where students are given a family such as y = ax + b, y = Aex + Be−x, or circles with parameters, and must remove constants systematically.
  1. 1

    Enough for 1 mark

    Forming a differential equation means eliminating arbitrary constants from a given family of curves by differentiating enough times and combining the resulting equations.

  2. 2

    Add for 2 marks

    Example: For y = mx, where m is an arbitrary constant, differentiate: dy/dx = m. Since m = y/x from the original equation, dy/dx = y/x. Thus x(dy/dx) - y = 0 is the required differential equation.

  3. 3

    Add for 3 marks

    A family of curves may contain arbitrary constants such as a, b, or c.

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