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.
cair0
Messages
15
Reaction score
0
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
 
Physics news on Phys.org
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...
 
Thread 'Struggling to make relation between elastic force and height'
Hello guys this is what I tried so far. I used the UTS to calculate the force it needs when the rope tears. My idea was to make a relationship/ function that would give me the force depending on height. Yeah i couldnt find a way to solve it. I also thought about how I could use hooks law (how it was given to me in my script) with the thought of instead of having two part of a rope id have one singular rope from the middle to the top where I could find the difference in height. But the...
Thread 'Voltmeter readings for this circuit with switches'
TL;DR Summary: I would like to know the voltmeter readings on the two resistors separately in the picture in the following cases , When one of the keys is closed When both of them are opened (Knowing that the battery has negligible internal resistance) My thoughts for the first case , one of them must be 12 volt while the other is 0 The second case we'll I think both voltmeter readings should be 12 volt since they are both parallel to the battery and they involve the key within what the...
Back
Top