Homework Help: 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.