Introduction to Differential Equation

MathsDifferential EquationsFor JEE aspirants

A differential equation is an equation that links an independent variable, a dependent variable and one or more derivatives of the dependent variable, such as $\frac{d y}{d x} + x y = sin x$ . This introduction to differential equations explains ordinary and partial equations, how to read the order and degree, how to form the differential equation of a family of curves, and what general, particular and singular solutions mean. Order, degree and formation are standard questions in JEE Main, JEE Advanced and Class 12 boards.

On this page1What is a DE2Order and degree3Formation4Types of solution5Slope field6Solved examples

Key Formulas - Quick Reference

★ Must learnOrder = order of the highest derivative present (always a positive integer).
★ Must learnDegree = power of the highest order derivative, after the equation is made a polynomial in all its derivatives.
Degree is not defined when a derivative sits inside $sin$ , $e^{()}$ , $lo g$ and so on, e.g. $e^{y^{'''}} - x y^{''} + y = 0$ .
★ Must learnA family with $n$ essential arbitrary constants has a differential equation of order $n$ .
★ Must learnFormation: differentiate $n$ times, then eliminate all $n$ constants.
General solution: exactly as many arbitrary constants as the order. Particular solution: none left.
$y = m x$ (lines through $O$ ) $\Rightarrow x \frac{d y}{d x} = y$ ; all non-vertical lines $y = m x + c \Rightarrow \frac{d ^{2} y}{d x ^{2}} = 0$ .
First order, first degree form: $\frac{d y}{d x} = f (x, y)$ , or $M d x + N d y = 0$ .

1. What is a Differential Equation?

An equation involving independent and dependent variables and the derivatives of the dependent variable(s) is called a differential equation. The unknown in such an equation is not a number but a function: we look for $y = y (x)$ whose derivatives make the equation true. There are two kinds of differential equation: ordinary and partial.

Figure 1: In

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

the highest derivative is the second, so order

= 2

; its power is 2, so degree

= 2

. The cube on

\frac{d y}{d x}

never decides the degree.

1.1 Ordinary differential equation (ODE)

If the dependent variable(s) depend on only one independent variable

x

, the differential equation is called ordinary.

Examples:

$\frac{d y}{d x} + \frac{d z}{d x} = y + z$ (two dependent variables $y$ , $z$ , one independent variable $x$ )
$\frac{d y}{d x} + x y = sin x$
$\frac{d ^{3} y}{d x ^{3}} + 2 \frac{d y}{d x} + y = e^{x}$
$k \frac{d ^{2} y}{d x ^{2}} = {1 + (\frac{d y}{d x})^{2}}^{3/2}$
$y = x \frac{d y}{d x} + k 1 + (\frac{d y}{d x})^{2}$

1.2 Partial differential equation (PDE)

If the dependent variable depends on two or more independent variables, the equation contains partial derivatives and is called a partial differential equation.

Examples: $y^{2} \frac{\partial z}{\partial x} + y \frac{\partial ^{2} z}{\partial y ^{2}} = a x$ and $\frac{\partial ^{2} z}{\partial x ^{2}} + \frac{\partial ^{2} z}{\partial y ^{2}} = 0$ . JEE and board questions use only ordinary differential equations, so from here on "differential equation" means ODE.

Ordinary (ODE)

One independent variable.
Ordinary derivatives $\frac{d y}{d x}$ , $\frac{d ^{2} y}{d x ^{2}}$ .
Solution: a curve $y = y (x)$ .

Partial (PDE)

Two or more independent variables.
Partial derivatives $\frac{\partial z}{\partial x}$ , $\frac{\partial z}{\partial y}$ .
Solution: a surface $z = z (x, y)$ .

Figure 2: Count the independent variables first: one gives an ODE, two or more give a PDE. JEE and board questions use ODEs only.

Key idea

Count the independent variables: one means ODE. The unknown of a differential equation is a whole function, not a number.

2. Order and Degree of a Differential Equation

2.1 Order

Order is the order of the highest derivative appearing in the differential equation. It is always a positive integer.

In $\frac{d ^{3} y}{d x ^{3}} + 2 \frac{d y}{d x} + y = e^{x}$ the highest derivative is the third, so the order is 3. Powers do not matter for order: $(\frac{d y}{d x})^{5} = y$ is still of order 1.

2.2 Degree

Degree is the highest power (exponent) of the highest order derivative after the equation is cleared of radicals and fractions as far as the derivatives are concerned.

Note: for the degree to exist, every derivative must appear in polynomial form. An equation of the form

f_{1} (x, y) [\frac{d ^{m} y}{d x ^{m}}]^{n_{1}} + f_{2} (x, y) [\frac{d ^{m - 1} y}{d x ^{m - 1}}]^{n_{2}} + \dots + f_{k} (x, y) [\frac{d y}{d x}]^{n_{k}} = 0

has order $m$ and degree $n_{1}$ . The coefficients $f_{1}, f_{2}, \dots$ may be any functions of $x$ and $y$ ; only the derivatives need to appear as whole-number powers.

Find the highest derivative: its order is the order of the equation.
If any derivative is inside a transcendental function ( $sin$ , $cos$ , $e^{()}$ , $lo g$ , $tan^{- 1}$ ...), stop: the degree is not defined.
Remove fractional powers and fractions in the derivatives (raise both sides to a power, multiply through).
Read the power of the highest derivative: that is the degree.

Figure 3: Order first, then the two degree checks. Example:

e^{y^{'''}} - x y^{''} + y = 0

stops at the first diamond (order 3, degree not defined).

Equation	Order	Degree	Reason
$\frac{d y}{d x} + x y = sin x$	1	1	already a polynomial in $y^{'}$
$\frac{d ^{3} y}{d x ^{3}} + 2 \frac{d y}{d x} + y = e^{x}$	3	1	$e^{x}$ has no derivative inside
$k \frac{d ^{2} y}{d x ^{2}} = {1 + (\frac{d y}{d x})^{2}}^{3/2}$	2	2	square both sides: $k^{2} (y^{''})^{2} = (1 + y^{'2})^{3}$
$y = x \frac{d y}{d x} + k 1 + (\frac{d y}{d x})^{2}$	1	2	$(y - x y^{'})^{2} = k^{2} (1 + y^{'2})$
$e^{y^{'''}} - x y^{''} + y = 0$	3	not defined	$y^{'''}$ inside an exponential
$\frac{d ^{2} y}{d x ^{2}} = sin y$	2	1	$sin y$ is fine: no derivative inside

Exam Trick

Fractional powers: multiply every exponent by the LCM of the denominators. For $(y^{'''})^{2/3} = (1 + y^{'})^{1/2}$ the LCM of 3 and 2 is 6, giving $(y^{'''})^{4} = (1 + y^{'})^{3}$ : order 3, degree 4. If an option says the degree is a fraction, it is wrong: degree is always a positive integer when it exists.

Only derivatives inside a function spoil the degree.

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

has degree 1, but

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

has no degree. When one whole side is a single function of the derivatives, undo it first:

y = e^{y^{'} + y^{''}}

becomes

y^{''} + y^{'} = ln y

, which has degree 1 (Solved Example 1).

Order

Which derivative is highest?
Counts differentiations.
Always exists.

Degree

What power is that derivative raised to?
Read only after clearing radicals.
May be not defined.

Quick Recall: tap to check

Order and degree of

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

Order 1, degree 3.

Does

y^{''} + e^{y} = 0

have a degree?

Yes, degree 1. The exponential holds

y

, not a derivative.

Order and degree of

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

Square:

(y^{''})^{2} = 1 + (y^{'})^{2}

. Order 2, degree 2.

Can the order of a differential equation be 0?

No. An equation with no derivative is not a differential equation; order is at least 1.

Key idea

Order comes from the highest derivative; degree is its power, and only after the equation is a polynomial in the derivatives.

3. Formation of a Differential Equation

A family of curves such as $y = m x$ (all lines through the origin) has one equation for every value of its arbitrary constant. The differential equation of the family is a single equation, free of the constant, that every member satisfies. It has two properties:

(a) its order is exactly the number of essential arbitrary constants in the equation of the curve;
(b) it contains no arbitrary constant.

3.1 Steps

Identify the number $n$ of essential arbitrary constants in the equation of the curve.
Differentiate the equation of the curve $n$ times.
Eliminate the $n$ constants between the equation of the curve and the $n$ new equations.

Note (essential constants): if arbitrary constants are joined by addition, subtraction, multiplication or division, club them into one new constant before counting.

y = a e^{x + b} = (a e^{b}) e^{x} = A e^{x}

has one essential constant;

y = (a + b) x

has one;

y = A cos x + B sin x

has two.

Figure 4: The order of the formed equation equals the number of essential constants. Merge constants that combine (sum, product) before counting.

Figure 4 runs the steps on $y = m x$ (Solved Example 2). Figure 5 draws that family of lines, and Figure 6 another one-constant family: circles touching the $x$ -axis at the origin (Solved Example 3).

Figure 5: One arbitrary constant

m

, so one differentiation:

y^{'} = m = \frac{y}{x}

. Every line of the family satisfies

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

(Solved Example 2).

Figure 6: One parameter

λ

fixes the size and side of the circle, so the family needs a first order equation:

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

(Solved Example 3).

Family of curves	Constants	Differential equation
$y = m x$ (lines through $O$ )	1	$x \frac{d y}{d x} = y$
$y = m x + c$ (all non-vertical lines)	2	$\frac{d ^{2} y}{d x ^{2}} = 0$
$x^{2} + y^{2} = r^{2}$ (circles centred at $O$ )	1	$x + y \frac{d y}{d x} = 0$
$y = A e^{k x}$ ( $k$ fixed)	1	$\frac{d y}{d x} = k y$
$y = A e^{x} + B e^{- x}$	2	$\frac{d ^{2} y}{d x ^{2}} = y$
$x^{2} + y^{2} + λ y = 0$ (circles touching $x$ -axis at $O$ )	1	$\frac{d y}{d x} = \frac{2 x y}{x ^{2} - y ^{2}}$

The parabolas with focus at the origin and axis along the $x$ -axis, $y^{2} = 4 a (x + a)$ , also form a one-constant family (Practice Question 6). Figure 7 shows a surprise: two members opening in opposite directions always cut at right angles.

Figure 7: One constant

a

, one differentiation:

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

. Members opening in opposite directions cut at

9 0^{\circ}

, e.g. at

(0, 2)

the slopes are

1

and

- 1

Exam Trick

Order = number of essential constants. Before forming anything, count the constants: a two-constant family must give a second order equation, so options of order 1 or 3 drop out at once. Then check a surviving option by putting one easy member (for example $m = 1$ ) into it.

JEE Advanced

All circles in the plane, $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ , have three essential constants, so their differential equation is of order 3:

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

It says the radius of curvature $\frac{( 1 + y ^{'2} ) ^{3/2}}{∣ y ^{''} ∣}$ is constant along each curve. Differentiating that ratio and setting the derivative to zero gives the equation above. The family in Figure 7 is self-orthogonal: replacing $\frac{d y}{d x}$ by $- \frac{d x}{d y}$ leaves its equation unchanged.

Key idea

Count essential constants, differentiate that many times, eliminate them: the order of the result equals the count.

4. Solution of a Differential Equation

Finding the dependent variable from the differential equation is called solving or integrating it. A solution (or integral) is a relation between the dependent and independent variables, free from derivatives, that satisfies the equation.

Note: the solution is also called a primitive, because the differential equation can be regarded as a relation derived from it (by differentiating and eliminating constants, exactly as in Section 3).

4.1 Three types of solution

Type	Meaning	Example for $y = x \frac{d y}{d x} - (\frac{d y}{d x})^{2}$
General (complete integral, complete primitive)	contains exactly as many independent arbitrary constants as the order	$y = c x - c^{2}$
Particular	obtained by giving values to the arbitrary constants of the general solution	$y = 2 x - 4$ (taking $c = 2$ )
Singular	satisfies the equation but cannot be obtained from the general solution for any value of the constants	$y = \frac{x ^{2}}{4}$

4.2 General and particular solutions

For $\frac{d y}{d x} = 2 x$ the general solution is $y = x^{2} + C$ : one constant for a first order equation. A condition such as $y (1) = 3$ (an initial condition) picks one member: $3 = 1 + C$ , so $C = 2$ and $y = x^{2} + 2$ .

Figure 8:

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

has general solution

y = x^{2} + C

(one constant, first order). The initial condition

y (1) = 3

gives

C = 2

, the particular solution

y = x^{2} + 2

(bold); dashed curves are other members.

Initial value problem: a differential equation of order

n

together with $n$ conditions (values of

y, y^{'}, \dots

at given points). The

n

conditions fix the

n

constants of the general solution.

Verifying a solution. To check that a function is a solution, differentiate it and substitute. For $y = e^{- 3 x}$ and $\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} - 6 y = 0$ : $y^{'} = - 3 e^{- 3 x}$ , $y^{''} = 9 e^{- 3 x}$ , and $9 e^{- 3 x} - 3 e^{- 3 x} - 6 e^{- 3 x} = 0$ . It is a solution, but a particular one: it has no arbitrary constant, while the general solution of this second order equation needs two (Solved Example 7).

General solution

Family of curves.
Number of arbitrary constants = order.
Example: $y = x^{2} + C$ .

Particular solution

One curve of the family.
Constants fixed by given conditions.
Example: $y = x^{2} + 2$ .

4.3 Singular solution

A singular solution is not obtainable from the general solution. Geometrically, the singular solution is the envelope of the family of curves represented by the general solution: a curve that touches every member. In Figure 9, every line $y = c x - c^{2}$ solves $y = x p - p^{2}$ (where $p = \frac{d y}{d x}$ ), and so does the parabola $y = \frac{x ^{2}}{4}$ they all touch. Singular solutions arise in equations of higher degree, such as Clairaut's equation $y = p x + f (p)$ , solved in the next concept.

Figure 9: For

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

every grey line

y = c x - c^{2}

is a solution, and so is the parabola

y = \frac{x ^{2}}{4}

that touches each line at

(2 c, c^{2})

. No value of

c

gives it: it is singular.

4.4 Geometric meaning: the direction field

A first order equation $\frac{d y}{d x} = f (x, y)$ tells you the slope of the solution curve at every point. Draw a short dash with that slope at many points and the solution curves appear as the paths that follow the dashes. For $\frac{d y}{d x} = x - y$ the general solution is $y = x - 1 + C e^{- x}$ ; every member approaches the line $y = x - 1$ .

Figure 10: A first order equation

\frac{d y}{d x} = f (x, y)

fixes a slope at every point (short dashes). Each solution curve

y = x - 1 + C e^{- x}

runs along these slopes; through each point passes exactly one member.

Exam Trick

Test options by differentiating, not by solving. If a question offers four candidate solutions, differentiate each and substitute into the equation. A wrong option usually fails at a simple point such as $x = 0$ within seconds.

Quick Recall: tap to check

How many arbitrary constants does the general solution of a third order equation have?

Exactly 3.

How many conditions fix a particular solution of a second order equation?

Two, e.g.

y (0)

and

y^{'} (0)

y = 0

a particular solution of

y = x p - p^{2}

Yes: it is the member

c = 0

y = c x - c^{2}

. The parabola

y = \frac{x ^{2}}{4}

is the singular one.

Key idea

General solution = whole family (constants = order); particular = one member picked by conditions; singular = the envelope no constant can give.

5. First Order, First Degree Equations

Most equations in JEE and board papers are of first order and first degree: $\frac{d y}{d x} = f (x, y)$ , often written as $M d x + N d y = 0$ with $M$ , $N$ functions of $x$ and $y$ . Their solving methods (variables separable, homogeneous, exact, linear, Bernoulli) are the next concept on this site.

Figure 11: The whole concept on one screen. Order and degree classify, formation builds the equation, and the three kinds of solution undo it.

6. Solved Examples

Solved Example 1

Find the order and degree of the following differential equations.
(i)

\frac{d ^{2} y}{d x ^{2}} = [y + (\frac{d y}{d x})^{6}]^{1/4}

(ii)

y = e^{y^{'} + y^{''}}

(iii)

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

(iv)

e^{y^{'''}} - x y^{''} + y = 0

Solution:

(i) Raise both sides to the power 4:

(\frac{d ^{2} y}{d x ^{2}})^{4} = y + (\frac{d y}{d x})^{6}

Highest derivative $y^{''}$ , power 4: order 2, degree 4.

(ii) Take $ln$ of both sides: $\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = ln y$ . Order 2, degree 1.

(iii) Take $sin^{- 1}$ of both sides: $\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = sin^{- 1} y$ . Order 2, degree 1.

(iv) $y^{'''}$ sits inside an exponential, so the equation cannot be written as a polynomial in the derivatives. Order 3, degree not defined.

Solved Example 2

Form the differential equation of the family of straight lines passing through the origin.

Solution:

The family is $y = m x$ , where $m$ is a parameter (one constant).
Differentiate once with respect to $x$ : $\frac{d y}{d x} = m$ .
Eliminate $m$ using $m = \frac{y}{x}$ .

Answer: $\frac{d y}{d x} = \frac{y}{x}$ , i.e. $x \frac{d y}{d x} = y$ (Figure 5).

Solved Example 3

Form the differential equation of the family of circles touching the

x

-axis at the origin.

Solution:

Such a circle has centre $(0, k)$ and radius $∣ k ∣$ : $x^{2} + y^{2} + λ y = 0$ ... (i), where $λ$ is a parameter.
Differentiate: $2 x + 2 y \frac{d y}{d x} + λ \frac{d y}{d x} = 0$ ... (ii).
From (i), $λ = - \frac{x ^{2} + y ^{2}}{y}$ . Put it in (ii) and multiply by $y$ : $2 x y + (2 y^{2} - x^{2} - y^{2}) \frac{d y}{d x} = 0$ .

Answer: $\frac{d y}{d x} = \frac{2 x y}{x ^{2} - y ^{2}}$ (Figure 6).

Solved Example 4

The order and degree of

(\frac{d ^{3} y}{d x ^{3}})^{2/3} = (1 + \frac{d y}{d x})^{1/2}

are respectively
(A)

3, 2/3

(B)

3, 4

(C)

3, 2

(D)

1, 3

Solution:

The LCM of the denominators 3 and 2 is 6. Raising both sides to the power 6 gives $(\frac{d ^{3} y}{d x ^{3}})^{4} = (1 + \frac{d y}{d x})^{3}$ , a polynomial in the derivatives. Highest derivative: third, power 4.

Answer: (B). Option (A) reads the degree before clearing the radical; a degree is never a fraction.

Solved Example 5

The differential equation of all non-vertical lines in the

x y

-plane is
(A)

d y / d x = 0

(B)

d^{2} y / d x^{2} = 0

(C)

d x / d y = 0

(D)

d^{2} x / d y^{2} = 0

Solution:

Non-vertical lines: $y = m x + c$ , two essential constants, so the equation is of order 2.
$\frac{d y}{d x} = m$ , then $\frac{d ^{2} y}{d x ^{2}} = 0$ : both constants are gone.

Answer: (B). Option (A) describes only horizontal lines, and (D) is the equation of all non-horizontal lines $x = a y + b$ .

Solved Example 6

Form the differential equation of the family

y = a e^{x + b}

, where

a

and

b

are arbitrary constants, and state its order.

Solution:

Club the constants: $y = (a e^{b}) e^{x} = A e^{x}$ . Only one essential constant.
Differentiate once: $\frac{d y}{d x} = A e^{x} = y$ .

Answer: $\frac{d y}{d x} = y$ , order 1 (not 2, although two letters appear).

Solved Example 7

Verify that

y = c_{1} e^{2 x} + c_{2} e^{- 3 x}

is the general solution of

\frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} - 6 y = 0

, and find the particular solution with

y (0) = 0

and

y^{'} (0) = 5

Solution:

$y^{'} = 2 c_{1} e^{2 x} - 3 c_{2} e^{- 3 x}$ , $y^{''} = 4 c_{1} e^{2 x} + 9 c_{2} e^{- 3 x}$ .
$y^{''} + y^{'} - 6 y = (4 + 2 - 6) c_{1} e^{2 x} + (9 - 3 - 6) c_{2} e^{- 3 x} = 0$ . It is a solution with two independent constants for a second order equation, so it is the general solution.
$y (0) = c_{1} + c_{2} = 0$ and $y^{'} (0) = 2 c_{1} - 3 c_{2} = 5$ give $c_{1} = 1$ , $c_{2} = - 1$ .

Answer: $y = e^{2 x} - e^{- 3 x}$ .

Solved Example 8

Form the differential equation of all circles of fixed radius

r

whose centres lie on the

x

-axis. State its order and degree.

Solution:

Family: $(x - a)^{2} + y^{2} = r^{2}$ , with $a$ arbitrary ( $r$ is fixed): one constant.
Differentiate: $2 (x - a) + 2 y \frac{d y}{d x} = 0$ , so $x - a = - y \frac{d y}{d x}$ .
Substitute back: $y^{2} (\frac{d y}{d x})^{2} + y^{2} = r^{2}$ .

Answer: $y^{2} [1 + (\frac{d y}{d x})^{2}] = r^{2}$ ; order 1, degree 2.

Practice Questions

Find the order and degree: $\frac{d y}{d x} + y = \frac{1}{d y / d x}$ .Answer: order 1, degree 2 (multiply by $\frac{d y}{d x}$ : $(\frac{d y}{d x})^{2} + y \frac{d y}{d x} = 1$ ).
Find the order and degree: $e^{y^{'} - y^{'''}} = ln (\frac{d ^{5} y}{d x ^{5}} + 1)$ .Answer: order 5, degree not defined.
Find the order and degree: $[(\frac{d y}{d x})^{1/2} + y]^{2} = \frac{d ^{2} y}{d x ^{2}}$ .Answer: order 2, degree 2.
Obtain the differential equation of the family $y = a sin (b x + c)$ , where $a$ and $c$ are arbitrary constants ( $b$ fixed).Answer: $\frac{d ^{2} y}{d x ^{2}} + b^{2} y = 0$ .
Show that the differential equation of the system of parabolas $y^{2} = 4 a (x - b)$ is $y \frac{d ^{2} y}{d x ^{2}} + (\frac{d y}{d x})^{2} = 0$ .Answer: $2 y y^{'} = 4 a$ ; differentiating again, $2 y^{'2} + 2 y y^{''} = 0$ .
Form the differential equation of the family of parabolas with focus at the origin and axis of symmetry along the $x$ -axis.Answer: $y^{2} = y^{2} (\frac{d y}{d x})^{2} + 2 x y \frac{d y}{d x}$ (from $y^{2} = 4 a (x + a)$ ).
Form the differential equation of $y = A cos 2 x + B sin 2 x$ .Answer: $\frac{d ^{2} y}{d x ^{2}} + 4 y = 0$ .

Common Mistakes to Avoid

Watch out

Reading the degree before clearing radicals: $\frac{d ^{2} y}{d x ^{2}} = 1 + (y^{'})^{2}$ has degree 2, not 1 (square both sides first).
Giving a degree when a derivative is inside $e^{()}$ , $sin$ , $lo g$ : for $e^{y^{'''}} - x y^{''} + y = 0$ the degree is not defined.
Taking the largest power anywhere as the degree. In Figure 1 the cube on $\frac{d y}{d x}$ does not count; only the power of the highest derivative does.
Thinking $sin y$ or $e^{x}$ in an equation kills the degree. Only derivatives inside such functions matter: $y^{''} + sin y = 0$ has degree 1.
Counting letters instead of essential constants: $y = a e^{x + b}$ gives a first order equation.
Leaving an arbitrary constant in the formed equation, or differentiating fewer times than the number of constants.
Calling the singular solution a particular solution. A particular solution comes from the general solution; a singular one never does.
Using fewer conditions than the order to fix a particular solution: a second order equation needs two.

Frequently Asked Questions

What is a differential equation?

A differential equation is an equation that contains an unknown function and one or more of its derivatives, such as $\frac{d y}{d x} + x y = sin x$ . Solving it means finding the function, usually as a family of curves with arbitrary constants, that makes the equation true for all x in an interval.

What is the difference between the order and degree of a differential equation?

The order is the order of the highest derivative present, for example 2 when the second derivative is the highest. The degree is the power of that highest derivative after the equation has been made a polynomial in all its derivatives by clearing radicals and fractions.

When is the degree of a differential equation not defined?

The degree is not defined when the equation cannot be written as a polynomial in its derivatives, for example when a derivative appears inside an exponential, sine, logarithm or inverse trigonometric function, as in $e^{(y^{'''})} - x y^{''} + y = 0$ . The order is still defined in such cases.

How do you form a differential equation from a family of curves?

Count the essential arbitrary constants in the equation of the family, say n. Differentiate the equation n times and eliminate all n constants between the n + 1 equations. The result is a differential equation of order n with no arbitrary constants.

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

The general solution contains as many independent arbitrary constants as the order and represents a whole family of curves. A particular solution is one member of that family, obtained by giving fixed values to the constants, usually from initial conditions such as y(0) = 1.

What is a singular solution of a differential equation?

A singular solution satisfies the differential equation but cannot be obtained from the general solution for any value of its constants. Geometrically it is the envelope of the family given by the general solution, such as the parabola $y = \frac{x ^{2}}{4}$ touching every line $y = c x - c^{2}$ .

Which differential equation topics are in the JEE Main syllabus?

The NTA syllabus for JEE Main lists ordinary differential equations with their order and degree, solution by separation of variables, and homogeneous and linear equations $\frac{d y}{d x} + P (x) y = Q (x)$ . Formation of equations from curve families and orthogonal trajectories also appear in JEE Advanced level practice.

Why does the number of arbitrary constants equal the order of the differential equation?

Each differentiation produces one new equation, and eliminating n independent constants needs n extra equations. So a family with n essential constants needs exactly n differentiations, and its differential equation has order n. Conversely, solving an order n equation brings back n constants.

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