Having trouble with differential equations and separation of variables

[cair0]

two problems in particular, one i got in calc, the other in physics

one asks if a = -v
and v = 1 when t = 0
what is a possible position function for this equation


the other one is
given a = 3x
and starting at rest from x = 0
find the velocity at 5 seconds

i can't seem to get the concept behind these, because the times we do them are so far and few between

 

[gnome]

For the first one:

since a = dv/dt = -v, you could solve it as a separable equation by writing it as
dv/v = -dt
and integrating both sides.

But you should really be able to do this one just by inspection.

Start off by thinking of a function that is equal to its own derivative & then think of how you can modify it to be equal to the negative of its derivative.

If you need more of a clue, look at the last item on this page:
https://www.physicsforums.com/showthread.php?t=4463&page=1&pp=15
(I can't understand why chroot didn't like it; I loved it. )

Then give it a constant coefficient C and use the given boundary condition v(0) = 1 to find the value of C.

 

[HallsofIvy]

I started to do some complicated calculations on the second question when suddenly it hit me: the objects acceleration is proportional to x and x= 0? And its initial speed is also 0?? What does that tell you?

 

[cair0]

I started to do some complicated calculations on the second question when suddenly it hit me: the objects acceleration is proportional to x and x= 0? And its initial speed is also 0?? What does that tell you?

the assumption is that it will accelerate...

 

[cair0]

For the first one:

since a = dv/dt = -v, you could solve it as a separable equation by writing it as
dv/v = -dt
and integrating both sides.

But you should really be able to do this one just by inspection.

Start off by thinking of a function that is equal to its own derivative & then think of how you can modify it to be equal to the negative of its derivative.

If you need more of a clue, look at the last item on this page:
https://www.physicsforums.com/showthread.php?t=4463&page=1&pp=15
(I can't understand why chroot didn't like it; I loved it. )

Then give it a constant coefficient C and use the given boundary condition v(0) = 1 to find the value of C.

yeah that one was really obvious now that i think about it, for some reason i kept getting stuck with the 2nd derrivative of x = the 1st derivative of x, and that notation ws getting me nowhere...