**Differential Equation**

An equation consisting of one (or more) independent variables and one (or more) dependent variables and **at least one derivative** is known as a differential equation (D.E.).

The derivatives can be ordinary or partial. Depending on them, a D.E. can be ordinary D.E. or partial D.E.

[Note the development so far: Set Theory, Relations, Functions, Limit of a Function, Derivatives, Integration, Differential Equations]

**Order and Degree**

Order of a D.E. is the order of the **highest ordered derivative** in the D.E. Degree of a D.E. is the power to which the highest ordered derivative is raised, provided that D.E. is free from radical and no derivative is present in the denominator.

In this section, we will deal with D.E. of 1 independent variable, or , and 1 dependent variable .

**Solution of a Differential Equation**

The relation , which satisfies the given D.E. is known as the solution of that D.E. For example, is a solution of

and are arbitrary constants. The solution containing arbitrary constants is known as a **general solution**. If values of arbitrary constants are given, it becomes a **particular solution**.

**Formation of a D.E.**

It can be seen that the degree of differential equation is equal to the number of arbitrary constants in its solution. So, to form a D.E. from its **solution** with arbitrary constants, we need to differentiate it times. The arbitrary constants are then to be eliminated.

**Types of Ordinary D.E. of 1st order and 1st degree**

**Variable-Separable Form**

The variables can be integrated separately. D.E. can be written in the form

**Illustration** : Solve the D.E. (**June 2015**)

**Solution** : The terms containing and those containing can be separated. Hence, this is a variable-separable type D.E.

Therefore,

This is the required solution.

**Homogeneous Equations**

If the sum of powers of and in each term of a D.E. is same, it is said to be **homogeneous**. A homogeneous D.E. can be solved by substituting

Illustration : Solve the D.E.

**Non-homogeneous Equations**

These are of the form:

**Type 1**

When , write .

It then reduces to

Writing as , we get a variable-separable form in and .

**Type 2**

When , write .

Find and such that and

The equation then becomes

This is a homogeneous D.E. and can be solved by substitution .

**Exact (or Total) Differential Equations**

An equation is said to be exact, when the expression containing differentials and can be reduced to a single differential . The solution of this D.E. will then be .

For to be exact, check for

The equation can then be solved by:

or

**Illustration** : Solve the D.E. (**June 2016**)

**Solution** : Cross-multiplying,

Comparing this with , we get and .

Since and are equal, this D.E. is exact. The solution is

This gives

On integrating,

Or

Note that this is a general second degree equation, which may represent various conics, depending on the value of .

**Tip for MCQs** : Any differential equation of the form will be exact if .

**D.E.s Reducible to Exact D.E.**

An **integrating factor** is the term, with which if a D.E. is multiplied, it becomes exact. Since it is easy to solve an exact D.E., more attention is given to develop methods to find the integrating factor.

If the equation is homogeneous and , then integrating factor is .

If the equation is of the form , then the integrating factor is .

If, for the given D.E., then the integrating factor is $e^{\int f(x)dx}$.

If, for the given D.E. & $ e^{\int f(y)dy}$

If the given D.E. is of the form , then latex \gamma$ and by testing the condition of exactness.

**Linear D.E.**

When the order of a D.E. is 1, it is linear.

The standard form is

The solution is given by

and are functions of only.

**Bernoulli’s Equation**

This equation is of the form

It can be reduced to linear form by dividing it by and then substituting as . The equation then becomes a linear equation in and .

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