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).

One constant $c$ → differentiate once: $\dfrac{dy}{dx} = 2cx$ .
From the original, $c = \dfrac{y}{x^2}$ . Substitute: $\dfrac{dy}{dx} = 2\cdot\dfrac{y}{x^2}\cdot x = \dfrac{2y}{x}$ .
So $x\dfrac{dy}{dx} = 2y$ .

Answer:

x\dfrac{dy}{dx} - 2y = 0

Practice this conceptself-check · 3 quick reps

Try it yourself

Form the differential equation of the family of parabolas

x^2 = 4ay

with vertex at the origin.

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
How many times to differentiate a 2-constant family?
2.
ODE of $y = A - \frac{B}{x}$ (two constants)?
3.
ODE of $y = e^x(a\cos x + b\sin x)$ ?

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.

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?

Separate: $by\,dy = ax\,dx$ , integrate: $\dfrac{b}{2}y^2 = \dfrac{a}{2}x^2 + C$ .
Rearrange: $\dfrac{a}{2}x^2 - \dfrac{b}{2}y^2 + C = 0$ .
A circle needs equal coefficients on $x^2$ and $y^2$ : $\dfrac{a}{2} = \dfrac{b}{2}$ with opposite signs after moving terms, i.e. $a = -b$ .

Answer:

a = -b

(and nonzero).

Practice this conceptself-check · 3 quick reps

Try it yourself

The general solution of

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

is a circle only when?

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
To confirm a family solves a given ODE, you?
2.
Integrated $x^2 - y^2 = c$ — circle or hyperbola?
3.
Equal, same-sign $x^2$ and $y^2$ coefficients give a?

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 (2)

Differentiate as many times as there are constants
→ Eliminating arbitrary constants
A circle needs equal squared-term coefficients
→ Matching an ODE to its general solution

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Differential EquationsHARD

The differential equation of the system of circles touching the y-axis at the origin is

[Q89 · Apr · 2019]

Example 2Differential EquationsEASY

Which one of the following differential equations has the general solution

y=ae^x+be^{-x}

[Q98 · Apr · 2021]

Example 3Differential EquationsHARD

The differential equation representing the curve

y=e^x(a\cos x+b\sin x)

, where

a

and

b

are arbitrary constants, is

[Q72 · Apr · 2024]

Example 4Differential EquationsEASY

The differential equation of the family of curves

y=p\cos(ax)+q\sin(ax)

, where p, q are arbitrary constants, is

[Q87 · Sep · 2018]

Example 5Differential EquationsMODERATE

What is the differential equation of all parabolas of the type

y^2=4a(x-b)

[Q70 · Sep · 2023]

Drill every past-year question on this subtopic

12 questions from the bank — paginated, with cart and Word-export support.

Drill the 12 questions