Classification of First order Differential Equation

Classification of First order Differential Equation
A standard form for first order diff. eq . on the unknown function y(x) , y^ =f(x,y)
Can be written as differential form : M(x,y)dx+N(x,y)dy=0 ……(1)
Or as: dy/dx=M(x,y)/-N(x,y)
Where M(x,y),N(x,y) are functions
Where the differential forms may be infinite for same eq. in standard form .
Example : The eq. in standard form as; xy^ -y^2=0
From differential forms ; xdy=y^2 dx
dy/dx=y^2/x
To could be solved ,and the classifications is :
Separable Equation.:
The standard form (1) above if can written as:
M(x,y)=A(x) function on x
N(x,y)=B(y) function on y
and A(x)dx+B(y)dy=0
Then the diff. eq. is separable eq. ,The solution of separable :
??A(x) dx+??B(y) dy=c
Where c is arbitrary constant .
The integration may be numerical so we can approximate solution. Initial value problem
A(x)dx+B(y)dy=0 y(x_0 )=y_0
Can be solved as
?_(x_0)^x?A(x) dx+?_(y_0)^y?B(y) dy=0
Then g(x)dx+f(y)dy=c



Example : Solve y^ =y^2 x^3
Rewrite as separable eq. as; x^3 dx-(1/y^2 )dy=0 as A(x)dx+B(y)dy=0
Solve this eq. ; ???x^3 dx?-??(1/y^2 ) dy=c ?(?? ) x^4/4+1/y=c
Solving explicitly for y ,the solution will be as;
y=(-4)/(x^4+k)
Where k=-4c
Example : Consider the equation
which is defined for all . Since the right hand side vanish for y = 0, the constant function y = 0 is a solution.
In the domains y > 0 and y < 0, the equation can be solved using separation of variables. For example, in the domain y > 0, then ;


whence

and

(the restriction x > -C comes from the previous line). Similarly, in the domain y < 0,
we obtain

whence

and

We obtain the following integrals curves: