Few beginner doubts about differential equations ?

[kupid]

I was trying to picture the third derivative of something
Then i came across these ...

What does displacement mean?

The variable x is often used to represent the horizontal position. The variable y is often used to represent the vertical position

Displacement=Delta x=xf-x0xf refers to the value of the final position.
x0 refers to the value of the initial position
Delta x is the symbol used to represent displacement



Distance = Speed x Time
Time = Distance/Speed
Speed= Distance/Time

Average velocity is defined to be the change in position divided by the time of travel

Acceleration is defined to be the rate of change of the velocity

Jerk is defined to be the rate of change of acceleration
What is the term used for the third derivative of position?

It is well known that the first derivative of position (symbol x) with respect to time is velocity (symbol v) and the

second is acceleration (symbol a). It is a little less well known that the third derivative, i.e. the rate of change of

acceleration, is technically known as jerk (symbol j). Jerk is a vector but may also be used loosely as a scalar quantity

because there is not a separate term for the magnitude of jerk analogous to speed for magnitude of velocity.

In the UK jolt has sometimes been used instead of jerk and may be equally acceptable
The more derivative you take the steeper the slope gets ?

Also , when you integrate a " jerk " , You get acceleration ?
Integrate acceleration to get velocity ?
Integrate velocity to get displacement ?So , when a differential equation contains the third derivative of something like that or a " jerk ", we can integrate it to find the original displacement function ?
I was not able to find an example for a third order differential equation containing a "jerk"
This is the only example i got ...

Looks like something interesting to follow .

Separation of variables is a technique commonly used to solve first-order ordinary differential equations. It is so-called because we rearrange the equation to be solved such that all terms involving the dependent variable appear on one side of the equation, and all terms involving the independent variable appear on the other. Integration completes the solution. Not all first-order equations can be rearranged in this way so this technique is not always appropriate. Further, it is not always possible to perform the integration even if the variables are separable. In this Section you will learn how to decide whether the method is appropriate, and how to apply it in such cases

http://www.personal.soton.ac.uk/jav/soton/HELM/workbooks/workbook_19/19_2_first_order_odes.pdf

https://www.khanacademy.org/math/calculus-home
https://www.khanacademy.org/math/differential-equationsPlease help ...

 

[MarkFL]

Let's talk about the simple case of motion in one dimension, i.e. along a straight line. A particle's position at time $t$ is commonly given as:

\(\displaystyle s(t)\)

Velocity $v$ is defined as the time rate of change of position:

\(\displaystyle v(t)=\d{s}{t}\)

Acceleration $a$ is defined as the time rate of change of velocity:

\(\displaystyle a(t)=\d{v}{t}=\d{^2s}{t^2}\)

Jerk $j$ is defined as the time rate of change of acceleration:

\(\displaystyle j(t)=\d{a}{t}=\d{^2v}{t^2}=\d{^3s}{t^3}\)

Now, when integrating for example to go from acceleration to velocity, we will need to know one point on the velocity curve in order to determine the constant of integration. We are typically given an initial value, and it is this initial value along with the differential equation that constitutes an initial value problem (IVP). If we are simply told:

\(\displaystyle \d{v}{t}=2\)

Then we find there is a family of curves that satisfy the given ordinary differential equation (ODE), namely:

\(\displaystyle v(t)=2t+c_1\)

If we denote the initial velocity as $v(0)=v_0$

Then our velocity becomes one curve, which is:

\(\displaystyle v(t)=2t+v_0\)

We see now that this function satisfies both the ODE and the initial condition, and so this is the solution to the IVP.

Before we proceed to your other questions, do you have any questions about what I just posted?

 

[kupid]

Thanks for the awesome reply Mark ,
When you add a little bit of physics these differential equation questions seems a little bit more interesting ,You get a physical picture of something we are trying to solve .

Nop , No more questions from Post #2

Please continue :)

 

[MarkFL]

Okay, your next question appears to be:

The more derivative you take the steeper the slope gets ?

I'm not sure what you mean by that...could you elaborate?

 

[kupid]

Maybe i missed this definition The slope of the curve (derivative) at a given point is a number.

Derivatives are found all over science and math, and are a measure of how one variable changes with respect to another variable
I guess the change in graph simply depends on the number or variable that changes with respect to another variable .?

 

[kupid]

Hey mark , I have been looking at post #2 for some time now trying to figure out a few things you mentioned in that post .

The problem is i am a person who is self teaching math and i am not familiar with all the mathematical notations and jargon .

This is my familiar view of functions .

Functions are pretty simple things , they just express a relationship between two different quantities

In a strict mathematics sense, y is just a variable. When someone writes "y=f(x)", it means that the value of y depends on the value of x, which is another variable. That is, for different values of x, there is a function, called f(x), which determines the value of y.

The x variable is therefore called the "independent" variable, while the y variable is called the "dependent" variable because it's value "depends" on the value of x

Now to a little bit of unfamiliar way of viewing functions

is it possible to rewrite the above equations in terms of f(x) = something ? y(x) = something ? or y=f(x) ?

 

[MarkFL]

Typically in kinematics, position, velocity, and acceleration are functions of time and as such $t$ is used as the independent variable simply for clarity. But, you can label your variables anything you want. :D

 

[kupid]

Thanks a lot Mark ,

Can i rearrange a few things in this thread for one last time , it is suddenly looking like a nice list of things to follow ...

Functions are pretty simple things , they just express a relationship between two different quantities

In a strict mathematics sense, y is just a variable. When someone writes "y=f(x)", it means that the value of y depends on the value of x, which is another variable. That is, for different values of x, there is a function, called f(x), which determines the value of y.

The x variable is therefore called the "independent" variable, while the y variable is called the "dependent" variable because it's value "depends" on the value of x

Now to a little bit of unfamiliar way of viewing functions

Is it possible to rewrite the above equations in terms of f(x) = something ? y(x) = something ? or y=f(x) ?

Typically in kinematics, position, velocity, and acceleration are functions of time and as such t is used as the independent variable simply for clarity. But, you can label your variables anything you want .

https://math.dartmouth.edu/archive/m23f04/public_html/Terminology.pdf

Separation of variables is a technique commonly used to solve first-order ordinary differential equations. It is so-called because we rearrange the equation to be solved such that all terms involving the dependent variable appear on one side of the equation, and all terms involving the independent variable appear on the other. Integration completes the solution. Not all first-order equations can be rearranged in this way so this technique is not always appropriate. Further, it is not always possible to perform the integration even if the variables are separable. In this Section you will learn how to decide whether the method is appropriate, and how to apply it in such cases

http://www.personal.soton.ac.uk/jav/soton/HELM/workbooks/workbook_19/19_2_first_order_odes.pdf
https://www.khanacademy.org/math/calculus-home
https://www.khanacademy.org/math/differential-equations

Please feel free to delete it if i have messed up the thread

 

[kupid]

Let me add this site too for anyone struggling with math .

17-University Precalculus
17Calculus - You CAN Ace Calculus
17Calculus - Differential Equations

Looks really nice for beginners