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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete 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.

Definition

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

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.

Rule to remember

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.

Memory hook

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

Examples and method

Worked example

Example: Form the differential equation of y = ax + b. Differentiate once: dy/dx = a. Differentiate again: d2y/dx2 = 0. Since both arbitrary constants are removed, the required differential equation is d2y/dx2 = 0.

Method to apply

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

Diagram support

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

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 = Ae^x + Be^(-x), or circles with parameters, and must remove constants systematically.

Avoid common mistakes

Common confusion

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.

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.

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.

Quick check

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

Twice, because the family contains two arbitrary constants, a and b.

Answer writing and exam use

1-mark answer

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

2-mark answer

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

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. 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. Example: Form the differential equation of y = ax + b. Differentiate once: dy/dx = a. Differentiate again: d2y/dx2 = 0. Since both arbitrary constants are removed, the required differential equation is d2y/dx2 = 0. Asked as a short-answer or long-answer item where students are given a family such as y = ax + b, y = Ae^x + Be^(-x), or circles with parameters, and must remove constants systematically. Stopping after dy/dx = a for y = ax + b, which still contains the arbitrary constant a and therefore is not the required differential equation.
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