### Equations of First Order

**In this post we will discuss about the equation of first order but not first degree .It is usually denoted**

**dy/dx**by

**p .**

Thee are three types of such equations

1) Equations solvable for

**p**.

2)Equation solvable for

**y .**

3)Equation solvable for

**x**.

**Equation Solvable for p :-**examples like this

**p.p +2py cot x = y.y**and its solution is

**[y-(c/1+cos x)][y-(c/1-cos x)] = 0**

is the equation solvable for

**p**.

**Equation Solvable for y :-**Let in the given differential equation , on solving for

**y**, given that ;

**y=f(x,p) ---------------(1)**

Differentiating with respect to

**x**, we obtain ;

**p=dy/dx=A(x,p,dp/dx)**

so that we obtain a new differential equation with variables

**x**and

**p**.

Suppose that it is possible to solve the equation

Let the solution be

**F(x,p,c)=0 ----------(2)**

where

**c**is the arbitrary constant .

The equation of

**(1)**may be exhibited in either of the two forms . We may either eliminate

**p**between

**(1)**and

**(2)**and obtain

**A(x,y,c)**as the required solution or we may solve

**(1)**and

**(2)**for

**x , y**and obtain .

**x=f'(p,c) and y=f"(p,c)**

as required solution where

**p**is the parameter .

**Equations Solvable for x :-**Let the given differential equation , on solving for

**x**, gives

**x=f(p,y) -----------------------(1)**

differentiating with respect to

**y**we obtain

**1/p=dy/dx=A(y,p,dp/dx)**; say

So that we obtain a new differential equation in variables

**y**and

**p**, Suppose that it is possible to solve the equation .

Let the solution be

**F(p,y,c)=0 -------------------(2)**

After the elimination

**p**between

**(1)**and

**(2)**will give the solution . Express

**x**and

**y**in terms of

**p**and

**c**where

**p**is to be regarded as parameter .

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