Solution To First Order DE Problem - Eliminating Dependent Variable (Solving For x)

 Solve by finding a general solution: 

py ² -  x = 0  

Re-arrange to express x alone: 

x=   py ² 

Replace p with dy/dx:

y ²  (dy/dx)  -  x  = 0

Separate variables:

y ²  (dy/dx)  =  x  

Set up for integration:

y ²  ò  dy  =   x ò dx

Yielding:

  y ³ /3  =   x ² / 2  + c

®

 x ²    =  2 y ³ /3  - 2C

Let  C =  -2c

(New arbitrary constant)

Solving for x to get general soln.

x  =  + Ö  2 y ³ /3  +  c