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ExtravagantDreams

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My intent is to create a thread for people interested in Differential Equations. However, I will explicitly state that I am only a student of this class myself and that many things could end up being incorrect or an improper way to present the material.

I will merely be going along with the class, mostly using excerpts and questions from the book, "Elementary Differential Equations and Boundary Value Problems: Seventh Edition," by William E. Boyce and Richard C. DiPrima. So truthfully, this is more for myself. Looking things up and explaining it to others seems to be the best way to learn.

If people have any questions or comments, feel free to share. Also, I know there are many knowledgeable people on this board, so be sure to correct me or make suggestions.

This will require knowledge of Calculus but don't be shy to ask if there is something that you are unsure of.

First, a little background;

What is a Differential Equation?

A Differential Equation is simply an equation containing a derivative.

Classifications:

Ordinary Differential Equations (ODE) - Equations that appear with ordinary derivatives (single independent variable, could have multiple dependent variables).

Examples:

[tex]

\frac {dy} {dt} = ay - b

[/tex]

[tex]

a \frac {dy_1} {dx} + b \frac {dy_2} {dx} + cy_1 = dy_2 = e

[/tex]

Partial Differential Equations (PDE) - Equations that appear with partial derivatives (multiple independent variable).

Examples:

[tex]

\alpha^2 [ \frac {\partial^2 u(x,t)} {\partial x^2} ] = \frac {\partial u(x,t)} {\partial t}

[/tex]

[tex]

\frac {\partial^2 V(x,y)} {\partial x^2} + \frac {\partial^2 V(x,y)} {\partial y^2} = 0

[/tex]

Don't let any of this frighten you. Math is always scary when looked at a glance with a bunch of undefined variables.

Linear and Nonlinear

The ordinary differential equation:

[tex]

F(t, y, y', ..., y^{(n)}) = 0

[/tex]

is said to be linear if F is a linear function of the variables y, y',..., y

[tex]

a_0 (t) y^{(n)} + a_1 (t) y^{(n-1)} + ... + a_n (t) y = g(t)

[/tex]

where

An example of a simple Nonlinear ODE would simply be:

[tex]

y \frac {dy} {dx} = x^4

[/tex]

This concludes the introduction. I may or may not write the next chapter tonight. However, a question, does anyone know an easier way for writing math on the computer and one that looks less confusing. I know I will have difficulty finding some things, especially subscripts and superscripts. Anyone know a better way to denote these?

I will merely be going along with the class, mostly using excerpts and questions from the book, "Elementary Differential Equations and Boundary Value Problems: Seventh Edition," by William E. Boyce and Richard C. DiPrima. So truthfully, this is more for myself. Looking things up and explaining it to others seems to be the best way to learn.

If people have any questions or comments, feel free to share. Also, I know there are many knowledgeable people on this board, so be sure to correct me or make suggestions.

This will require knowledge of Calculus but don't be shy to ask if there is something that you are unsure of.

First, a little background;

What is a Differential Equation?

A Differential Equation is simply an equation containing a derivative.

Classifications:

Ordinary Differential Equations (ODE) - Equations that appear with ordinary derivatives (single independent variable, could have multiple dependent variables).

Examples:

[tex]

\frac {dy} {dt} = ay - b

[/tex]

[tex]

a \frac {dy_1} {dx} + b \frac {dy_2} {dx} + cy_1 = dy_2 = e

[/tex]

Partial Differential Equations (PDE) - Equations that appear with partial derivatives (multiple independent variable).

Examples:

[tex]

\alpha^2 [ \frac {\partial^2 u(x,t)} {\partial x^2} ] = \frac {\partial u(x,t)} {\partial t}

[/tex]

[tex]

\frac {\partial^2 V(x,y)} {\partial x^2} + \frac {\partial^2 V(x,y)} {\partial y^2} = 0

[/tex]

Don't let any of this frighten you. Math is always scary when looked at a glance with a bunch of undefined variables.

Linear and Nonlinear

The ordinary differential equation:

[tex]

F(t, y, y', ..., y^{(n)}) = 0

[/tex]

is said to be linear if F is a linear function of the variables y, y',..., y

^{n}(Dependant variable must be first order). Thus the general linear ordinary differential equation of order n is:[tex]

a_0 (t) y^{(n)} + a_1 (t) y^{(n-1)} + ... + a_n (t) y = g(t)

[/tex]

where

^{(n)}is not the power of but the nth derivative.An example of a simple Nonlinear ODE would simply be:

[tex]

y \frac {dy} {dx} = x^4

[/tex]

This concludes the introduction. I may or may not write the next chapter tonight. However, a question, does anyone know an easier way for writing math on the computer and one that looks less confusing. I know I will have difficulty finding some things, especially subscripts and superscripts. Anyone know a better way to denote these?

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