What is ordinary and partial differential equations? What are the difference between them? How do you solve th

Question

What is ordinary and partial differential equations?

What are the difference between them?

How do you solve them?

Do both oridinary and partial difference equations have first and second order?

How are ordinary or partial differential equations used in real life?

Please try to explain clearly with examples and anything related to help understanding the concepts easier.

Answer 1

A differential equation is simply an equation with a differential term in it, such as "dy/dx". Ordinary differential equations (or "ODEs", or as we called them in college "Diffy-Qs") only deal with one independent variable. Partial differentials (PDs) are derivatives of a function of more than one variable, but taken with respect to just one variable while the rest are assumed constant. ODEs typically use the letter "d", while PDEs use the greek letter δ (lower case delta).

For example: y(x) = 5(dy/dx) + 7x is an ODE. But something like A(x,y) = δA/δx + δA/δy would be a PDE. And yes, you can certainly have second and higher orders of both, as you take more and more derivatives.

As for how these are solved, there are entire classes dedicated to the subject. Engineering colleges usually require students take the class. Sometimes they're simple to solve if you know calculus. For example, "y = dy/dx" would mean "the function equal to its derivative". In this case, you could have something like y = Ce^x (where C is a constant). Solving problems of radio active half-lifes or populations based on birth & death rates are usually the first DEs math students see, in calculus class. There are some cases though where DEs can get really complicated, and you need to learn new tricks for solving them.

Usually ODEs and PDEs are suppled with additional information, such as "y(0) = 5". This is called an "initial condition". This is because you usually you have to do some integration where you get things like "C" (constants) you have to solve for, and plugging in the initial condition helps you solve for these.

As for how they're used in real life, there are countless engineering problems that require the use of DEs. Sometimes you're presented with a certain situation in chemistry, physics, or biology, and you want to write a set of equations that fit what you see (a process called "Mathematical Modeling"). Whenever you need to describe a system whose state is depending on how fast certian things are changing, you'll need to explain things in terms of derivatives, which means you'll need to write out and solve DEs.

Answer 2

The big difference between them is that ordinary differential equations contain complete derivatives whereas partial differential equations may also contain derivatives with respect to only one variable. Both types of differential equation can contain 1'st and 2'nd (or even higher) order derivative terms.

As for the rest of your question, there simply isn't enough room here to answer it. Why not take a 1 year course in ordinary and partial differential equations? Pretty much any University with a graduate program offering Doctoral degrees im math will have such a course in their catalog (altho, it may not be offered every year)


Doug

Answer 3

ordinary differntial equatin has only one independent variable.
but partial differentiation has a number of independent variables, (example x and y) so if u differentiate with respect to x its called partial differentiation with respect to x.

solving method:

in ordinary differentiation, if we differentiate with respect to one variable (say x) , then we consider the other variable as something dependent on x (i.e) (dy/dx)
in partial, both are considered as variables

both have second order

consider the eqn
y^2=2x+4y

ordinary diff eqn:
differentiate with respect to x
then 2ydy/dx=2 + 4 dy/dx
with respect to y
then the eqn becomes
2y=2dx/dy+4




partial differentiation:

both are independent variables

we don say 'd' here , we say 'doe' instead of 'd'
lets take eqn xy = 4


in partial diff ur answer is

partial diff
xdy/dx+ydx/dy=4

but if its ordinary diff eqn its., uv method, that is x dy/dx+y=4


note the difference
both are considered as independent variables in p.d


hope u understand the difference.. nice question

Answer 4

ordinary dr. when the fn. dep. on one var. and u diff. the fn. w.r.t. that
partial when the fn. dep. on more than 1 var. and u diff the fn. w.r.t. any of the var.
there can be 1st and higher order der. for both.
lots of diff. methods to solve . go to wolframsmathworld or so
just search for diff. eq. cannot explain everything here. lots of details.

Answer 5

not exactly... ode is much simpler than pde... with ode... you only have one variable... with pde.. there are several variables to consider... somehow... with numerical analysis... the ideas are not that far anyway... but with rigorous analysis for the closed form... they might diverge... §