Solving Boat Speed & Distance with Drag Force

[Macykc2]

Homework Statement

A boat with initial speed v_o is launched on a lake. The Boat is slowed by the water by a force F=-αe^βv. Find an expression for the speed v(t), and find the time and distance for the boat to stop.

Homework Equations

Drag force F=-αe^βv
alpha and beta are not specified as to what they represent.

The Attempt at a Solution

I have an attached attempted solution, I'm just kind of confused on my results, I feel like it makes sense but at the same time it doesn't. My confusion lies in that I try to imagine using my derived v(t) equation but the drag force is constantly changed based on the instantaneous velocity at any moment, the v in the force equation is what throws me off, and makes me think I need to use a different approach.
Once I found my v(t) I found time when v(t)=0, then I found velocity as a function of position to find distance when v=0.
But again, now that I look at it, the v that is apart of e should probably be 0 as well, then the positions and times won't be affected by the drag force...

 

[haruspex]

Homework Statement

A boat with initial speed v_o is launched on a lake. The Boat is slowed by the water by a force F=-αe^βv. Find an expression for the speed v(t), and find the time and distance for the boat to stop.

Homework Equations

Drag force F=-αe^βv
alpha and beta are not specified as to what they represent.

The Attempt at a Solution

I have an attached attempted solution, I'm just kind of confused on my results, I feel like it makes sense but at the same time it doesn't. My confusion lies in that I try to imagine using my derived v(t) equation but the drag force is constantly changed based on the instantaneous velocity at any moment, the v in the force equation is what throws me off, and makes me think I need to use a different approach.
Once I found my v(t) I found time when v(t)=0, then I found velocity as a function of position to find distance when v=0.
But again, now that I look at it, the v that is apart of e should probably be 0 as well, then the positions and times won't be affected by the drag force...

Posting working as an image makes it hard to comment on individual lines. (It can also be hard to read, but yours is ok.)
Better to take the trouble to type the equations in, but if you must post images please number all the equations.

You cannot integrate a function of v just by multiplying it by t. v is a variable.
Rearrange the equation so that all the references to v are on one side and t on the other (including dv, dt).

 

[Macykc2]

I figured that's what i had to do, I took a look at it after I posted and tried some other things including that, and although I do get results it's once again hard to see if it makes sense, as they do not specify what alpha and beta are.

Also I will try in the future to just use this site instead of a picture, I'm just not good with that math function as I've never used it.

I ended up with v(t)=$\frac{1}{β}$ln(e^βv_o - $\frac{αβt}{m})$ but as I said, it's hard to know if what I did is right with the lack of info.

 

[Macykc2]

I also just noticed a very crucial mistake, this whole time I've been writing e^βv as e^-βv, so I need to re-do everything...