12Diff eq.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
Example1: Solve the differential equation(x+2y) p^3+3(x+y) p^2+(y+2x)p=0
Solution : Analyzing the equation as;
p(p+1)[(x+2y)+(2x+y)=0
The equation of first coefficientp=0
i.e
dy/dx=0
theny+c=0
The equation of second coefficientp+1=0
i.e
dy/dx=-1
then
y+x+c=0
The equation of third coefficient(x+2y) dy/dx+2x+y=0
i.e
(xdy+ydx)+2ydy+2xdx=0
then
xy+y^2+x^2+c=0