MathematicsClass 12ICSE

How Eliminating Constants Turns a Family of Curves Into One Equation

Find the order and degree of a differential equation, form differential equations by eliminating arbitrary constants, tell general from particular solutions, and solve first-order equations by separating the variables.

What is a differential equation, and why does it matter?

A differential equation links an unknown function to its derivatives. It describes how things change — a cooling cup of tea, a growing population, a charging battery — and solving it recovers the quantity itself.

This part covers order and degree, forming equations from families of curves, general and particular solutions, and the variable separable method.

How do you find the order and degree of a differential equation?

The order of a differential equation is the order of the highest derivative in it, and its degree is the power of that highest derivative once the equation is a polynomial in its derivatives; if it cannot be made polynomial, the degree is not defined.

Worked examples:

-

\frac{d ^{2} y}{d x ^{2}} + 3 (\frac{d y}{d x})^{3} + y = 0

— order 2, degree 1
-

(\frac{d ^{3} y}{d x ^{3}})^{2} + \frac{d y}{d x} = x

— order 3, degree 2
-

\frac{d y}{d x} + sin (\frac{d y}{d x}) = 0

— order 1, degree not defined

Clearing a radical first. For

\frac{d ^{2} y}{d x ^{2}} = 1 + (\frac{d y}{d x})^{2}

, squaring gives

(\frac{d ^{2} y}{d x ^{2}})^{2} = 1 + (\frac{d y}{d x})^{2}

, so the order is 2 and the degree is 2.

An everyday example. A car's motion described through its acceleration leads to a second-order equation, because acceleration is the second derivative of position.

The substance. The degree comes from the highest derivative only — the cube on

\frac{d y}{d x}

in the first example does not change its degree of 1.

How do you form a differential equation by eliminating arbitrary constants from a family of curves?

To form the differential equation of a family with n arbitrary constants, differentiate its equation n times and eliminate the n constants from the resulting equations; the answer has order n.

Worked example (one constant). For

y = A e^{3 x}

, differentiating gives

\frac{d y}{d x} = 3 A e^{3 x} = 3 y

, so the equation is

\frac{d y}{d x} - 3 y = 0

.

Worked example (two constants). For

y = A cos 2 x + B sin 2 x

:

-

\frac{d y}{d x} = - 2 A sin 2 x + 2 B cos 2 x

\frac{d ^{2} y}{d x ^{2}} = - 4 A cos 2 x - 4 B sin 2 x = - 4 y

- The equation is

\frac{d ^{2} y}{d x ^{2}} + 4 y = 0

, of order 2

Worked example (circles). Circles through the origin with centres on the x-axis satisfy

x^{2} + y^{2} = 2 a x

. Differentiating,

2 x + 2 y \frac{d y}{d x} = 2 a

; substituting for 2a gives

x^{2} + y^{2} = x (2 x + 2 y \frac{d y}{d x})

, that is

y^{2} - x^{2} = 2 x y \frac{d y}{d x}

.

An everyday example. All the paths of balls thrown with different speeds from the same spot form a family of parabolas, described together by one differential equation.

The substance. The number of independent constants fixes the order — a family with two such constants always gives a second-order equation.

What is the difference between the general solution and a particular solution of a differential equation?

The general solution of a differential equation contains as many arbitrary constants as its order and describes a whole family of curves, while a particular solution is the single member picked out when given conditions fix those constants.

Worked example. Verify that

y = A e^{2 x}

is the general solution of

\frac{d y}{d x} = 2 y

, and find the particular solution with

y (0) = 3

.

-

\frac{d y}{d x} = 2 A e^{2 x} = 2 y

, so it works for every value of A
-

y (0) = A = 3

, so the particular solution is

y = 3 e^{2 x}

- At

x = 1

y = 3 e^{2} \approx 22.2

Worked example 2.

y = A cos 2 x + B sin 2 x

is the general solution of

y^{''} + 4 y = 0

. With

y (0) = 1

and

y^{'} (0) = 4

, we get

A = 1

and

2 B = 4

, so

y = cos 2 x + 2 sin 2 x

.

An everyday example. Every possible balance in savings accounts growing at the same interest rate is described by the general solution; one account, with its own starting deposit, is a particular solution.

The substance. A second-order equation needs two conditions — one condition alone leaves a constant undecided.

How do you solve a first-order differential equation by separating the variables?

**If a first-order equation can be written as

g (y) d y = f (x) d x

, integrate both sides to get

\int g (y) d y = \int f (x) d x + C

; equations of the form

\frac{d y}{d x} = f (a x + b y + c)

reduce to this form with the substitution

v = a x + b y + c

.

Worked example.** Solve

\frac{d y}{d x} = \frac{1 + y ^{2}}{1 + x ^{2}}

\frac{d y}{1 + y ^{2}} = \frac{d x}{1 + x ^{2}} \Rightarrow tan^{- 1} y = tan^{- 1} x + C

Worked example 2 (particular solution). Solve

\frac{d y}{d x} = 2 x y

with

y (0) = 5

. Separating,

\frac{d y}{y} = 2 x d x

, so

lo g ∣ y ∣ = x^{2} + C

and

y = A e^{x^{2}}

. The condition gives

y = 5 e^{x^{2}}

, which is about 13.6 at

x = 1

.

Worked example 3 (reducible). Solve

\frac{d y}{d x} = (x + y)^{2}

. Put

v = x + y

, so

\frac{d v}{d x} = 1 + v^{2}

and

tan^{- 1} v = x + C

, giving

x + y = tan (x + C)

.

An everyday example. A cup of chai cooling at a rate proportional to how much hotter it is than a 25 °C room obeys

\frac{d T}{d t} = - k (T - 25)

, which separates to give

T = 25 + A e^{- k t}

.

The substance. Dividing can lose a solution — dividing by y in

\frac{d y}{d x} = 2 x y

assumes

y \neq = 0

, yet

y = 0

is also a solution.

Exam tip

What earns full marks on order, degree and variable separable equations?

**Clear radicals before stating a degree, and in separable equations write the separated form

g (y) d y = f (x) d x

as its own line before integrating.

- Order: the highest derivative; degree: its power in polynomial form
- n arbitrary constants give an equation of order n
- The general solution has constants; a particular solution fixes them
- Add the constant C once, on one side

The trap.** Stating a degree for

\frac{d y}{d x} + e^{d y / d x} = 0

. It is not a polynomial in the derivative, so its degree is not defined.

Did you know

Why do populations with plenty of food start by growing exponentially?

When food and space are unlimited, a population grows at a rate proportional to its size:

\frac{d P}{d t} = k P

. Separating the variables gives

P = P_{0} e^{k t}

— exponential growth.

Real populations eventually run short of resources, so a better model slows growth as P approaches a limit K:

\frac{d P}{d t} = k P (1 - \frac{P}{K})

. This is still separable, and its solution is the S-shaped logistic curve.

The same equations describe the spread of news through a town and how quickly a new technology is taken up.

Exam relevance

How are order, degree and variable separable equations tested in JEE Main?

Differential Equations is a recurring JEE Main chapter, and variable separable equations are the starting point for every other method.

What gets asked. Order and degree of given equations, forming the equation of a family such as circles or parabolas, and particular solutions of separable equations evaluated at a point.

Question types. Multiple-choice and numerical-value questions, often asking for y at a given x.

The trap that costs marks. Stating a degree for an equation that is not polynomial in its derivatives, such as one containing

sin (\frac{d y}{d x})

Key takeaways

What must you be able to do from this part?

- Order and degree: the highest derivative, and its power once the equation is polynomial in derivatives
- Forming equations: differentiate as many times as there are constants, then eliminate them
- General and particular solutions: a family of curves, and the one member fixed by given conditions
- Variable separable:

\int g (y) d y = \int f (x) d x + C

, with substitutions for reducible forms

Can you solve

\frac{d y}{d x} = \frac{y}{x}

with

y (1) = 2

What is a differential equation, and why does it matter?

How do you find the order and degree of a differential equation?

How do you form a differential equation by eliminating arbitrary constants from a family of curves?

What is the difference between the general solution and a particular solution of a differential equation?

How do you solve a first-order differential equation by separating the variables?

What earns full marks on order, degree and variable separable equations?

Why do populations with plenty of food start by growing exponentially?

How are order, degree and variable separable equations tested in JEE Main?

What must you be able to do from this part?

Related study notes

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