Thursday, September 8, 2022

Solutions To Clairaut Differential Equation Problems

 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:

pdp/  (p - 1) + dy = 0

=>  [p + 1 +  1/ (p-1)] dp + dy = 0  

Solving:  

½ p+ 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)




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