This differential equation gives a better approximation of a falling object due to gravity, can anyone please solve this? Either wolfram alpha is being a bitch or maybe this is impossible...
>>7964489
tungstenalpha.com
>>7964491
it's an actual site..
>>7964489
Have you tried with a change of variables?
>>7964489
wolfram spit out a fuckey answer
(y(x)*sqrt(c1-a/y(x))/c1 + log.... = (c2+x)^2
The trick to doing these kinds of integrations by hand is to multiply both sides by y'--then each is an easy derivative. After that the integration isn't entirely trivial, but is standard.
Just look up trajectories/orbits in central force problems. Conservations laws/etc. are more useful in the general case--as you might've guessed gravitational orbits are highly sensitive to initial conditions.
>>7964616
>The trick to doing these kinds of integrations by hand is to multiply both sides by y'--then each is an easy derivative.
What the actual fuck are you taking about?
Do you know what a second order differential equation is?
>>7964642
This equation can easily be solved the chain rule, you fucknut.
The result isn't even interesting op. All you get a conservative field like everybody fucking knows.
God fucking sci YOU'RE ALL FUCKING STOOPPEDDDDDDDDDDDDD
GET OFF MY BOARD
FUCKING NORMIES
REEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
>>7964642
>I am an idiot who says idiotic things so I assume everyone else does the same
>>7964663
Not Op, but I'm interested in how to solve this, care to explain a bit?
>>7964489
y double prime is the time derivative, yes?
So I get
y = (6 * G * M * t^2) ^ (1/4)
>>7964489
there is no hope
kill yourself
>>7964647
No need to be an asshole, I bet you were born knowing differential equations...
>>7964715
>taking the bait
Look d^2y/dt^2 = dv/dt = dv/dy x dy/dt = v dv/dy, seperate the variables and integrate normally.
.
>>7964489
just for you
>>7964489
>>7964798
pt 2
>>7964489
You will not find a nice solution for y(t). The 'correct' (read: less insane) way to approach this is, oddly enough, to solve for t(y) instead.
>>7964723 is how you would start. Eventually you are going to realise that this is really just equating the kinetic energy at any point during the fall to the loss of gravitational potential energy incurred so far from the fall. Along the way you should find yourself confronted with this expression [eqn]\mathrm{d}t = \frac{1}{\sqrt{2G}}\frac{\mathrm{d}y}{\sqrt{\tfrac{1}{y}-\tfrac{1}{y_0}}}[/eqn]This one is easily solved. Nonetheless, as you might have already noticed, solving this gives you the time as a function of y. Anyway if you are too lazy to perform the integration Wikipedia actually gives you the answer here https://en.wikipedia.org/wiki/Free_fall#Inverse-square_law_gravitational_field It also suggests that you recover y(t) from t(y) via Lagrange's inversion theorem, which in my opinion is not such a bad idea, given how futile Wolfram Alpha's efforts were at trying to solve your differential equation directly. The fact that y(t) comes out as an ugly series already suggests that solving directly for y(t) is not going to work in the first place.
It is worth pointing out that the problem encapsulated by your equation was first tackled by Newton under Problem 24 (Proposition 32) in the first book of the Principia. Newton likewise solved for the free-fall time t(y) instead of y(t) and he arrived at his result very elegantly in standard geometric fashion (without any calculus in fact). It is far more concise and beautiful than what is outlined above, and if I could draw efficiently on this board I would definitely be presenting his solution instead. I highly recommend reading his proof if you are sufficiently interested.
>>7964866
> muh newton's geometry proof is so beautiful
suicide is an option
>>7964489
>wanting a "better approximation"
>using newtonian gravity
>gravity is assumed to be the only force acting on the object
>drag exist, even in space
>G is constant
>accurate
Please let this be for studying/homework
>>7964919
>>>drag exist, even in space
>>7964921
>not knowing about solar wind
>>7964798
except the y(x) in the RHS of OPs problem was squared
>>7964921
>space is a perfect vacuum, cosmic dust is a myth
https://en.wikipedia.org/wiki/Cosmic_dust
>>7965433
This is neat, going to try the pool of water thing sometime to catch some space dust.
