Having trouble with differential equations and separation of variables

• Apr 20, 2004

• #1

cair0

15

0

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

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


the other one is
given [tex] a = 3x [/tex]
and starting at rest from [tex]x = 0[/tex]
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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• Apr 21, 2004

• #2

gnome

1,041

1

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.

 

• Apr 21, 2004

• #3

HallsofIvy

Science Advisor

Homework Helper

42,989

975

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?

 

• Apr 21, 2004

• #4

cair0

15

0

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...

 

• Apr 21, 2004

• #5

cair0

15

0

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. )

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...

 

1. What are differential equations?

Differential equations are mathematical equations that involve derivatives of an unknown function. They are used to describe relationships between different variables and their rates of change.

2. What is separation of variables?

Separation of variables is a method used to solve certain types of differential equations. It involves isolating variables on different sides of an equation in order to find a solution.

3. Why is it important to learn about differential equations and separation of variables?

Differential equations and separation of variables are important in many fields of science, engineering, and mathematics. They are used to model and solve real-world problems, such as predicting the growth of populations or the motion of objects.

4. What are some common challenges when working with differential equations and separation of variables?

Some common challenges include recognizing the type of differential equation and determining the appropriate method to solve it, as well as handling complex algebraic manipulations and dealing with boundary conditions.

5. How can I improve my understanding of differential equations and separation of variables?

Practice is key to improving your understanding of differential equations and separation of variables. It is also helpful to seek out additional resources, such as textbooks, online tutorials, and practice problems, and to seek guidance from a tutor or instructor if needed.