Differential Equations

I was solving this particular pair of differential equation.
Both \(x\) and \(y\) are function of \(t\)(time) \[10 \cos t-\dot{x}-\ddot{x}-\ddot{y}=0\]
10costx˙x¨(xy)=010 \cos t-\dot{x}-\ddot{x}-(x-y) =0
Where p˙\dot{p} and p¨\ddot{p} are single and double derivaties of pp with respect to tt (time) , respectively.

Is there any general form of solution, or some technique through Laplace transformation.
Any help will be appreciated.
Thanks in advance.

Note by Lil Doug
1 week, 2 days ago

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For something this complex, your best bet is probably numerical integration. Here is how it would look in Python (explicit Euler)

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x = x + xd*dt
y = y + yd*dt

xd = xd + xdd*dt
yd = yd + ydd*dt

xdd = 10.0*math.cos(t) - xd - (x-y)
ydd = 10.0*math.cos(t) - xd - xdd 

Steven Chase - 1 week, 2 days ago

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@Steven Chase Sir the code is not working

Lil Doug - 1 week, 2 days ago

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Yeah, it requires more code to actually run. I'll post the complete code soon

Steven Chase - 1 week, 2 days ago

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@Steven Chase @Steven Chase by the way I am very curious to know that how I will get xx and yy as a function of time because, python always gives results in numerical answer .
So I am very excited see that how python will do it?

Lil Doug - 1 week, 2 days ago

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@Lil Doug @Lil Doug Some differential equations can't be written as a function of time, like this one. This one is a nonlinear second order ODE; it is therefore extremely hard (or even impossible) to write a function of time of xx.

Take the pendulum with drag:

θ¨=gsin(θ)+Clθ˙\ddot{\theta} = -g \sin(\theta) + Cl\dot{\theta}

There are (as of now) no analytical solutions for the differential equation, as to the function θ\theta. The one you've posted above is much harder than the pendulum equation already.

Krishna Karthik - 1 week, 2 days ago

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@Steven Chase sir can you post a python based solution of my latest problem.
Thanks in advance.
Hope I am not disturbing you.

Lil Doug - 3 days, 20 hours ago

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Hello. It is up now

Steven Chase - 3 days, 20 hours ago

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Here is the full code, with some initialized values and plotting.

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import math

dt = 10.0**(-5.0)

##################################################

t = 0.0
count = 0

x = 1.0
y = 2.0

xd = -2.0
yd = 3.0

xdd = 10.0*math.cos(t) - xd - (x-y)
ydd = 10.0*math.cos(t) - xd - xdd

##################################################

while t <= 5.0:

    x = x + xd*dt
    y = y + yd*dt

    xd = xd + xdd*dt
    yd = yd + ydd*dt

    xdd = 10.0*math.cos(t) - xd - (x-y)
    ydd = 10.0*math.cos(t) - xd - xdd

    t = t + dt
    count = count + 1

    if count % 1000 == 0:
        print t,x,y

Steven Chase - 1 week, 2 days ago

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I want to know; some people do Explicit Euler in an inverted order; acceleration, velocity, then position, but you've done it in the opposite way. Is there a major difference in the two orders?

Krishna Karthik - 1 week, 2 days ago

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I doubt it makes too much difference. I basically taught myself how to do these things. So it wouldn't surprise me if my style was a bit unorthodox.

Steven Chase - 1 week, 2 days ago

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@Steven Chase @Steven Chase Do you use time-domain simulation as part of your day-to-day engineering?

Krishna Karthik - 1 week, 2 days ago

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@Krishna Karthik I do indeed. For simple things, I use little hand-crafted state-space simulations. For larger and more complex applications, we have more sophisticated (and much more expensive) time-domain simulation tools.

Steven Chase - 1 week, 2 days ago

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@Steven Chase @Steven Chase have a look on last 5 hour notifications

Lil Doug - 1 week, 2 days ago

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@Steven Chase How do you choose this values of x,x˙,y,y˙x, \dot{x}, y, \dot{y} are all these with random, or these value are basically a set, which are folloing the pair of differential equations.

Lil Doug - 1 week, 2 days ago

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You can initialize the position and velocity any way you want. And then the initial accelerations are determined by the differential equations.

Steven Chase - 1 week, 2 days ago

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@Steven Chase But how
x=5,x˙=6,y=7,y˙=8x=5, \dot{x}=6, y=7, \dot{y}=8
So these are my assumed values, now how to find acceleration?

Lil Doug - 1 week, 2 days ago

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@Lil Doug Re-arrange your second equation to solve for the x acceleration. Then plug that into your first equation and re-arrange to solve for the y acceleration.

Steven Chase - 1 week, 2 days ago

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@Lil Doug If you have any questions about Explicit Euler, ask me, if Steven Chase's busy.

Krishna Karthik - 1 week, 2 days ago

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I think he is lazy instead of busy.

Lil Doug - 1 week, 2 days ago

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@Lil Doug Lol; just ask me your question. If it's about differential equations or numerical solving, I'm sure I can answer it. I taught myself that stuff a while ago, like him.

Krishna Karthik - 1 week, 2 days ago

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@Krishna Karthik @Krishna Karthik Thanks
By the way, I have posted a mechanics problem, don't forget to solve and post a solution

Lil Doug - 1 week, 2 days ago

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@Steven Chase who said that the value of x is 1 ??
I am again very much curious to know your method

Lil Doug - 1 week, 2 days ago

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I just chose some values to initialize the simulation with

Steven Chase - 1 week, 2 days ago

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