In this problem, we are asked to find the complete primitive and singular solution of the given differential equation.


The given differential equation is xp^2-2yp 4x=0.


To solve this differential equation, we will use the method of separation of variables. We will first separate the variables and then integrate both sides to obtain the solution.


The given differential equation can be written as:

xp^2-2yp 4x=0

Dividing both sides by x, we get:

p^2 - 2yp/x = 0

Multiplying both sides by dx, we get:

p^2 dx - 2y/x dx = 0

Separating the variables, we get:

p^2 dx = 2y/x dx


Integrating both sides, we get:

∫p^2 dx = ∫2y/x dx

Integrating the left-hand side, we get:

x(p^2) = 2xy + C1

where C1 is the constant of integration.


Solving for p, we get:

p = ±√(2y/x + C1/x)

This is the complete primitive solution of the given differential equation.


To find the singular solution, we need to set the constant of integration C1 to zero. Therefore, the singular solution is given by:

p = ±√(2y/x)

This solution is singular because it does not include all possible solutions of the differential equation.


In this problem, we used the method of separation of variables to find the complete primitive and singular solution of the given differential equation. The complete primitive solution is given by p = ±√(2y/x + C1/x), and the singular solution is given by p = ±√(2y/x).