Separable First Order Differential Equation

[tinopham]

Homework Statement

$\frac{dy}{dx} = y \sqrt{x}$, f(9) = 5

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,$y = \frac{5}{e^{18}} e^{\frac{2}{3}x^\frac{3}{2}}$
$y = 5e^{-18} e^{\frac{2}{3}x^\frac{3}{2}}$
$y = 5 e^{\frac{2}{3}x^\frac{3}{2}-18}$

 

[kushan]

have ln y + ln c

 

[SammyS]

Homework Statement

$\frac{dy}{dx} = y \sqrt{x}$, f(9) = 5

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,$y = \frac{5}{e^{18}} e^{\frac{2}{3}x^\frac{3}{2}}$
$y = 5e^{-18} e^{\frac{2}{3}x^\frac{3}{2}}$
$y = 5 e^{\frac{2}{3}x^\frac{3}{2}-18}$

Hello tinopham. Welcome to PF !

Do you have a question about this ?

 

[tinopham]

Hi SamS, I was going to ask a question, but I was able to solve it. Thanks!

 

[HallsofIvy]

have ln y + ln c

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.