A relation between dependent variable and independent variables along with arbitrary constants that satisfy a differential equation is called its **solution**. The solutions of a differential equations are equations of curves.

If the number of arbitrary constants in the solution is equal to the order of the differential equation, the solution is called as the **general solution**.

If the arbitrary constants in the general solution are given particular values, the solution is called a **particular solution** (of the differential equation).

The order of differential equation corresponding to an equation having 'n' parameters is 'n'.

This is because we have to differentiate 'n' times successively to eliminate all the 'n' parameters

(say, from the equation of a given family of curves containing 'n' parameters).

**Example: ** The general solution of the differential equation y = ()x is y = mx.

Assigning a particular value to 'm' say – ,

y = (– )x is a particular solution of the differential equation.

## Solving differential equations

A general first order, first degree differential equation is of the form

= F(x, y)

Methods to solve "first order first degree" differential equations can be classified into:

i) Variables separable form

ii) Homogeneous differential equations

iii) Non-homogeneous differential equations

iv) Linear differential equations

A function f(x, y) of two variables x and y is said to be homogeneous function of degree 'α' if

f(kx, ky) = k^{α} f(x, y) for all values of 'k'.

f(x, y) can be written as

f(x, y) = x^{α} Φ(y/x)

Let us consider equations of the type

where f and g are homogeneous functions of the same degree in x and y.

Such equations are called **homogeneous equations**.

The above is a differential equation in 'v' and 'y'.

The two variables are separable and can be solved by variables separable method.

Finally, substitute back v = x/y to get the required solution.

## Case (ii):

Let a/a' = b/b' = m |

So ..... (1) |

Let ax + by = v ..... (2) |

Differentiating w.r.t. 'x' |

Substituting (2) & (3) in (1), |

This again can be solved by variables separable method. If b = – a' with (a/a') = (b/b') it is better to solve using case (i) rather than case (ii). |

## Case (iii):

b ≠ – a' and a/a' ≠ b/b' |

Let X, Y be variables and h, k be constants |

Let x = X + h and y = Y + k |

(Note the usage of small and capital letters). |

So we have |

Therefore becomes |

Choose constants such that |

ah + bk + c = 0 |

a'h + b'k + c' = 0 |

∴ |

The above is a homogeneous equation in X and Y. |

Hence homogeneous equation method is to be followed i.e., substitute Y = VX and proceed. |

A linear differential equation of n^{th} order is of the form

where P_{1}, P_{2}, ..... P_{n} and R are constants or functions of 'x' only.

What is its degree and order?

The dependent variable 'y' and its derivatives can be considered as raised to the power of 1.

Hence the degree is 1.

Hence it is also called as **linear differential equation**.

The highest derivative of 'y' is of order 'n'. Hence the order of the differential equation is 'n'.

We shall consider linear differential equations of first order only i.e., n = 1

A general form of such equation is

+ Py = Q

where P and Q are constants or functions of 'x' only.

To solve the above, first multiply both sides by e^{∫Pdx}

Using the product rule of differentiation at LHS, it becomes

Integrating the above we have,

which is the required solution.

e^{∫Pdx} is called the **integrating factor** which is denoted as **IF**.

So, y.IF = ∫(Q.IF)dx + c

Recall that linear differential equation of first order in 'x' is of the form

+ Px = Q

where P and Q are constants or functions of 'y' alone.

Now IF = e^{∫Pdy}
The solution is x.IF = ∫(Q.IF)dy + c

An equation of the form + Py = Qy^{n},

where P and Q are functions of 'x' only,

is called a Bernoulli's differential equation.

## Solution of Bernoulli's equation

Given equation is

It is linear differential equation in 'v'.

If M and N are functions of x and y,

the equation M dx + N dy = 0 is called exact differential equation

when there exists a function f(x, y) such that

d[f(x, y)] = M dx + N dy

i.e, dx + dy = M dx + N dy

where = partial derivative of f(x, y) w.r.t. 'x' (keeping 'y' constant)

= partial derivative of f(x, y) w.r.t. 'y' (keeping 'x' constant)

The necessary and sufficient condition for the differential equation

M dx + N dy = 0 to be exact is

= .

An exact differential equation can always be derived from its general solution directly by differentiation without any subsequent multiplication, elimination etc.

## Integrating factor

If an equation of the form M dx + N dy = 0 is not exact,

it can always be made exact by multiplying by some function of x and y.

Such a multiplier is called an **integrating factor**.

It is of the form:

y = x +
f()

Let = P ------- (1)

⇒ y = xP + f(P)

Differentiate above equation w.r.t. 'x'.

⇒

But = P

⇒ P =

⇒ dP/dx = 0 or x + (df/dP) = 0

P = constant (c)

So replace 'P' by 'c' in the differential equation.