text: # Separable First Order Differential Equation
  
  1. Feb 22, 2012
  
  ### tinopham
  
  1. The problem statement, all variables and given/known data
  
  $\frac{dy}{dx} = y \sqrt{x}$, f(9) = 5
  
  3. The attempt at a solution
  
  $\int dy/y = \int \sqrt{x} dx$
  
  $ln |y| = \frac{2}{3} x^\frac{3}{2} + c$
  
  $y = e^{\frac{2}{3}x^\frac{3}{2}} + C$
  
  $y = Ce^{\frac{2}{3}x^\frac{3}{2}}$
  
  $5 = Ce^{\frac{2}{3}9^\frac{3}{2}}$
  
  $5 = Ce^{18}$
  
  $C = \frac{5}{e^{18}}$
  
  Thus,$y = \frac{5}{e^{18}} e^{\frac{2}{3}x^\frac{3}{2}}$
  $y = 5e^{-18} e^{\frac{2}{3}x^\frac{3}{2}}$
  $y = 5 e^{\frac{2}{3}x^\frac{3}{2}-18}$
  
  Last edited: Feb 22, 2012
  2. Feb 22, 2012
  
  ### kushan
  
  have ln y + ln c
  
  3. Feb 22, 2012
  
  ### SammyS
  
  Staff Emeritus
  Hello tinopham. Welcome to PF !
  
  4. Feb 22, 2012
  
  ### tinopham
  
  Hi SamS, I was going to ask a question, but I was able to solve it. Thanks!
  
  5. Feb 22, 2012
  
  ### HallsofIvy
  
  1) The integral of 1/y is ln|y|+ c, not ln y.
  2) Since tinopham had a "c" on the right side ogf the equation, it is not necessary to have a constant on the left. The two constants of integration can be combined on oneside.