How to Solve a Differential Equation Using Separation of Variables

[bdh2991]

Homework Statement

use separation of variables to solve the differential equation x^2dy/dx=y-xy

with the initial condition of y(-1)=-1

Homework Equations

The Attempt at a Solution

after i separated and integrated i got the answer y=e^(-1/x-lnx+c)

the answer in the book is y=e^-(1+1/x)/x

i can't figure out how they got to that even after i plugged in -1 for x and y

some help would be greatly appreciated

 

[jamesrc]

You've done the calculus correctly, now you need to do a little algebra.

Hint: what's e^(-ln x)?

 

[SammyS]

Homework Statement

use separation of variables to solve the differential equation x^2dy/dx=y-xy

with the initial condition of y(-1)=-1

The Attempt at a Solution

after i separated and integrated i got the answer y=e^(-1/x-lnx+c)

the answer in the book is y=e^-(1+1/x)/x

i can't figure out how they got to that even after i plugged in -1 for x and y

some help would be greatly appreciated

$\displaystyle e^{-1/x-\ln(x)+C}=e^{-1/x}e^{-\ln|x|}e^C$

$\displaystyle =e^{-1/x}(1/x)e^C$​

Does that help?

 

[bdh2991]

well i think it would be x^-1 so then that would give me the x in the denominator then plugging in the initial values i should get c=1?

 

[SammyS]

well i think it would be x^-1 so then that would give me the x in the denominator then plugging in the initial values i should get c=1?

e⁰ = 1

 

[bdh2991]

ok thanks for the help i didn't remember that you could rewrite the e functions that way...