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)


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 : M(x,y)dx+N(x,y)dy=0 ………(1)
Or as: dy/dx=M(x,y)dx/-N(x,y)dy
Where M(x,y),N(x,y) are functions
To can be solved .and the classification is :
Separable Eq.:
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

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 : M(x,y)dx+N(x,y)dy=0 ………(1)
Or as: dy/dx=M(x,y)dx/-N(x,y)dy
Where M(x,y),N(x,y) are functions
To can be solved .and the classification is :
Separable Eq.:
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