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.