Differential Equations from First Order but High Degrees:

Differential Equations from First Order but High Degrees:

Divide into three kinds depending on solvability on one of its variables:
Case 1: Differential equation solvable on
p=dy/dx
If can be formed as ;

(p-G_1 (x,y))(p-G_2 (x,y))….(p-G_n (x,y))=0 ….(1)

Then each part after equaling with zero .formed a differential equation can be solved by one of previous methods
Suppose the solution of these equations as;
g_1 (x,y,c)=0 ,g_2 (x,y,c)=0 ,…….,g_n (x,y,c)=0
And the general solution for equation (1) is;
g_1 (x,y,c).g_2 (x,y,c)….g_n (x,y,c)=0
Note: the constants in equations that multiplied are equals since the equation (1) is from
first order
Case 2: Differential equation solvable on y ;
If can be solve the differential equation
F(x,y,p)=0…..(2)
For y
i.e
y=f(x,p)……(3)
Derive yforx ,and substituting on
p=dy/dx
Will be without y but with p,x and dp/dx,and solved forpbyx ;
Let the solution is ;
g(x,p,c)=0…..(4)
By deletingp from equations (3)and (4) we get on a relation for x,y and c, and thatis the solution .
Note: Sometimes there are additional solutions called particular solutions .