Ch 03/Formation of Differential Equations

Formation of Differential Equations

n arbitrary constants → differentiate n times → eliminate constants → DE of order n.

IThe Formation Rule

Every family of curves defined by n arbitrary constants gives rise to exactly one DE of order n. To find that DE:


Step 1: Write the equation of the family (with n constants)

Step 2: Differentiate n times to get n extra equations

Step 3: Eliminate all n constants between the original and the n derivative equations

Step 4: The result is the DE

Note

Think of it in reverse: a DE of order n has a general solution with n constants. Formation just goes the other direction — from the family of solutions to the DE that generates them.

Example 1 — One constanteasy

Form the DE from (C is arbitrary).

Show step-by-step solution+
Count constants
One constant C → will need to differentiate once
Differentiate
Find C from original
From :
Substitute C
DE
Example 2 — Two constantsmedium

Form the DE from (A, B arbitrary).

Show step-by-step solution+
Two constants → differentiate twice
1st derivative
2nd derivative
Observe that y'' = y
✓
DE
Example 3 — Circle familymedium

Form the DE for all circles centred at the origin: .

Show step-by-step solution+
One constant (r)
Differentiate implicitly
Simplify
DE
Geometric meaning
This says: at any point on the circle, the slope satisfies (tangent is perpendicular to radius).
Example 4 — Trigonometric familyhard

Form the DE from where b is a given constant.

Show step-by-step solution+
1st derivative
2nd derivative
Recognise y
DE
← Previous
Ch 2 · Classification of Differential Equations
Next chapter →
Ch 4 · Variable Separable Method
Take chapter quiz →