NDA Maths · Differential Equations

Forming an ODE from a Family of Curves

To form the differential equation of a family of curves, differentiate enough times to eliminate every arbitrary constant — n constants need n differentiations and produce an order-n equation.

All Differential Equations notes NDA Maths strategy

Why this matters

12 PYQs running the reverse of solving: you are given the answer (a family of curves) and must find the equation. The recipe never changes — differentiate, eliminate the constants — so these are dependable marks once the routine is automatic.

Concept 1 of 2

Eliminating arbitrary constants

Intuition

A family of curves with arbitrary constants is the general solution of some ODE. To recover that ODE, differentiate the family — each differentiation gives a new equation — until you have enough equations to eliminate every constant. With n constants, differentiate n times.

Definition

The elimination recipe:

Count the arbitrary constants — that is the order of the ODE you will get.
Differentiate the family that many times.
Eliminate the constants between the original equation and its derivatives; the constant-free relation is the ODE.
Examples: parabolas $x^2=4ay$ (one constant) → $x\,\dfrac{dy}{dx}=2y$ ; $y=e^x(a\cos x+b\sin x)$ (two constants) → $y''-2y'+2y=0$ .

Worked example

Form the differential equation of the family

y = cx^2

(c arbitrary).

Practice this conceptself-check · 3 quick reps

From the bank · past-year question

Example 1Differential EquationsMODERATE

What is the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis?

[Q99 · Sep · 2022]

Differentiate as many times as there are constants

A one-constant family needs one differentiation (order 1); a two-constant family like

y^2=4a(x-b)

needs two (order 2, giving

yy''+(y')^2=0

). Differentiating too few times leaves a constant stranded in the answer.

The order of the resulting ODE equals the number of constants

Before you differentiate, the family

y=Ae^{2x}+Be^{-3x}

has 2 arbitrary constants, so the eliminated ODE is order 2 — guaranteed. Reading the order off the highest derivative you happen to reach mid-working (or stopping early) gives the wrong order; count the independent constants first and that IS the order.

Drill 8 more on eliminating arbitrary constants

Concept 2 of 2

Matching an ODE to its general solution

Intuition

Sometimes you are handed both a candidate ODE and a family, and must check they correspond — or decide what condition makes a solution a particular shape (a circle, say). Either differentiate the family to confirm it fits the ODE, or solve the ODE and compare.

Definition

Two directions, one idea:

Family → ODE: differentiate and eliminate constants (as above), then compare with the given option.
ODE → family: integrate the separable ODE and read off the curve type.
A solved family is a circle only when the $x^2$ and $y^2$ coefficients are equal — e.g. $\dfrac{dy}{dx}=\dfrac{ax+h}{by+k}$ integrates to a circle exactly when $a=-b$ .

Worked example

For what relation between a and b does

\dfrac{dy}{dx} = \dfrac{ax}{by}

have circular solutions?

Practice this conceptself-check · 3 quick reps

From the bank · past-year question

Example 2Differential EquationsMODERATE

The general solution of

\dfrac{dy}{dx} = \dfrac{ax + h}{by + k}

represents a circle only when

[Q74 · Sep · 2017]

A circle needs equal squared-term coefficients

After integrating,

\frac{a}{2}x^2 - \frac{b}{2}y^2

is a circle only if those coefficients match in magnitude (giving

a=-b

); otherwise it is an ellipse or hyperbola. Don't assume any separable solution is a circle.

Drill 2 more on matching an ode to its general solution

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Watch out for (3)

Differentiate as many times as there are constants
→ Eliminating arbitrary constants
The order of the resulting ODE equals the number of constants
→ Eliminating arbitrary constants
A circle needs equal squared-term coefficients
→ Matching an ODE to its general solution

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