Finding general solution for ODE.

[Lengalicious]

Homework Statement

For the following differential equation:
1) Provide the general solution
2) Discuss for which values of x the solution is defined.
3) Find the solution of the initial value problem y(0) = 3

Homework Equations

dy/dx = (y+3)(y-5)

The Attempt at a Solution

1) so I separate variables and get (1/8)*(log(y-5)-log(y+3)) = x + C after integration.
Now i need to solve for x right? How would I go about doing this? Can't seem to figure out how to rearange.
2) I don't think i can do this until i find the general solution? But once its found do i just find its domain?
3) For this i just substitute the initial values in, solve for C and then place back in equation?

 

[Ray Vickson]

Homework Statement

For the following differential equation:
1) Provide the general solution
2) Discuss for which values of x the solution is defined.
3) Find the solution of the initial value problem y(0) = 3

Homework Equations

dy/dx = (y+3)(y-5)

The Attempt at a Solution

1) so I separate variables and get (1/8)*(log(x-5)-log(x+3)) = x + C after integration.
Now i need to solve for x right? How would I go about doing this? Can't seem to figure out how to rearange.
2) I don't think i can do this until i find the general solution? But once its found do i just find its domain?
3) For this i just substitute the initial values in, solve for C and then place back in equation?

How did you get ∫dy/[(y+3)(y-5)] = (1/8)[log(x-5)-log(x-3)]?

RGV

 

[Lengalicious]

sorry replace the x's with y's that was a typo. And i did it by splitting into partial fractions then integrating each term simply. Was that not correct?

 

[Ray Vickson]

Homework Statement

For the following differential equation:
1) Provide the general solution
2) Discuss for which values of x the solution is defined.
3) Find the solution of the initial value problem y(0) = 3

Homework Equations

dy/dx = (y+3)(y-5)

The Attempt at a Solution

1) so I separate variables and get (1/8)*(log(y-5)-log(y+3)) = x + C after integration.
Now i need to solve for x right? How would I go about doing this? Can't seem to figure out how to rearange.
2) I don't think i can do this until i find the general solution? But once its found do i just find its domain?
3) For this i just substitute the initial values in, solve for C and then place back in equation?

You don't need to solve for x; it is already done---just read your own equation! Now, if you wanted to solve for y as a function of x, that would involve a bit more effort, but is still just elementary algebra: use standard properties of log to do it.

RGV