Solutions To Clairault DE Problems Revisited

  1) Solve:  p² x - y  = 0

Solution:  

Solve for y:  

y =  p² x

Differentiate:

dy/dx = p =  p² + 2px (dp/dx)

Re-arrange:  

2 dp/(p -1) + dx/ x = 0  

Solve: 

(p - 1) ²  x = c  

Eliminate p between previous two eqns.  

y - 2Öx  + x =  c  

Which is the general soln.

2) Solve: p² + 2y  - 2x  = 0

Solution: 

Solve for x: 

x=  ½(2y + p²) 

Differentiate:

dx/dy = 1/p = 1  +  p (dp/dy)  

Simplify:

p² dp/  (p - 1) + dy = 0

=>  [p + 1 +  1/ (p-1)] dp + dy = 0  

Solving:  

½ p² + p  + ln (p - 1) + y = c

Which is the general solution.  The parametric form of the general solution is: 

y = c - ½ p²  - p  - ln (p - 1)

x = c  - p  - ln (p - 1)