Differential Equations

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

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

#Calculus

Easy Math Editor

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Comments

@Steven Chase @Karan Chatrath

Talulah Riley - 9 months ago

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

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 - 9 months ago

@Steven Chase Sir the code is not working

Talulah Riley - 9 months ago

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

Steven Chase - 9 months ago

@Steven Chase – @Steven Chase by the way I am very curious to know that how I will get $x$ and $y$ 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?

Talulah Riley - 9 months ago

@Talulah Riley – @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 $x$ .

Take the pendulum with drag:

$\ddot{\theta} = -g \sin(\theta) + Cl\dot{\theta}$

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

Krishna Karthik - 9 months ago

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

Talulah Riley - 8 months, 3 weeks ago

Hello. It is up now

Steven Chase - 8 months, 3 weeks ago

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

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 - 9 months ago

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 - 9 months ago

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 - 9 months ago

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

Krishna Karthik - 9 months ago

@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 - 9 months ago

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

Talulah Riley - 9 months ago

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

Talulah Riley - 9 months ago

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

Steven Chase - 9 months ago

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

Talulah Riley - 9 months ago

@Talulah Riley – 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 - 9 months ago

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

Krishna Karthik - 9 months ago

I think he is lazy instead of busy.

Talulah Riley - 9 months ago

@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 - 9 months ago

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

Talulah Riley - 9 months ago

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

Talulah Riley - 9 months ago

I just chose some values to initialize the simulation with

Steven Chase - 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$