Having trouble with differential equations and separation of variables

AI Thread Summary
The discussion focuses on solving two differential equations involving acceleration and velocity. For the first problem, where acceleration a = -v and initial velocity v(0) = 1, the solution involves recognizing that a function equal to its own derivative can be modified to fit the equation, leading to a separable equation dv/v = -dt. The second problem, with acceleration a = 3x and starting from rest at x = 0, highlights that the initial conditions imply no movement, suggesting a straightforward solution. Participants emphasize the importance of understanding the relationships between derivatives and initial conditions to simplify these problems. Overall, the thread illustrates the challenges and thought processes involved in applying separation of variables to differential equations.
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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
 
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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. :biggrin: )

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

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...
 
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