Can This Differential Equation Be Solved by Separation of Variables?

[SpaceIsCool]

Homework Statement

We solved the differential equation (2.29),

, for the velocity of an object falling through air, by inspection---a most respectable way of solving differential equations. Nevertheless, one would sometimes like a more systematic method, and here is one. Rewrite the equation in the “separated” form

and integrate both sides from time 0 to

to find

as a function of

. Compare with (2.30).

Homework Equations

[/B], equation to integrate/solve.

, answer should like this.

The Attempt at a Solution

Integrate both sides of the separated form of the differential equation from time 0 to t and then write

. After this, I will compare my answer to equation 2.30 which is

.

. This gives

. This is as far as I could get. I don't know how to get it into the form v_y(t).

 

[DrClaude]

Please read https://www.physicsforums.com/help/latexhelp/

 

[YoungPhysicist]

Please read https://www.physicsforums.com/help/latexhelp/

@SpaceIsCool In case you don’t understand that, your first relavent equation will look like:
$$(md\upsilon_y) / (\upsilon_y - \upsilon_{ter}) = -bdt$$

The code is:

(md\upsilon_y) / (\upsilon_y - \upsilon_{ter}) = -bdt

 

[SpaceIsCool]

@SpaceIsCool In case you don’t understand that, your first relavent equation will look like:
$$(md\upsilon_y) / (\upsilon_y - \upsilon_{ter}) = -bdt$$

The code is:

(md\upsilon_y) / (\upsilon_y - \upsilon_{ter}) = -bdt

Hello,

I'm not asking about latex help. Do I need to fix this before this question can be answered? I'm kinda new to using the forum.

 

[DrClaude]

I'm not asking about latex help. Do I need to fix this before this question can be answered? I'm kinda new to using the forum.

You can leave it as is for now, but please write the equations directly in the post instead of inserting them as figures. It makes the post harder to read and difficult to quote.

 

[SpaceIsCool]

You can leave it as is for now, but please write the equations directly in the post instead of inserting them as figures. It makes the post harder to read and difficult to quote.

Okay. I understand. Thanks!

 

[Chestermiller]

$v''=\frac{d^2v}{dt^2}$, not $\frac{dv}{dy}$. So your whole methodology from step 1 is incorrect.

If you had a course in differential equations, didn't they teach you how to solve an n'th order linear ordinary differential equation with constant coefficients?

 

[Delta2]

Perhaps the differential equation is $m\dot{v_y}=-b(v_y-v_{ter})$? if yes then what you doing seems correct except from the last integration step which should be

$$m(\ln|v_y-v_{ter}|-\ln|v_{y_0}-v_{ter}|)=-bt$$
This after some algebraic manipulation I believe it agrees with your equation (2.30).Otherwise if the ODE is exactly as you write at OP then as @Chestermiller notes, you go wrong from the first step cause it is $\ddot{v_y}=\frac{d^2v_y}{dt^2}$ and its NOT $\ddot{v_y}=\frac{dv_y}{dt}$ which you seem to use in order to do separation of variables.