Why Are Differential Equations Considered Difficult?

[Rockazella]

To start off I know nothing about them. My question is why are they considerd so difficult?

I'm in a precalc class and would like to study some higher level math on my own time. I have heard that these equations are very difficult to solve or understand or something, just wondering what makes this so?

 

[Tyger]

People differ greatly

in their ability to solve abstract problems. Most don't have much trouble with Algebra or Geometry, but for some reason Calculus seems to separate those with abstract abilites from those who don't.

 

[grady]

a differential equation is an equation with some kind of derivative in it. for instance dy/dx + y = 1. This is saying the function plus the first derivative of that same function is always equal to 1. There would be a family of y's that you could plug into this equation and that would be called the solution of the differential equation. Sometimes finding a solution can be difficult be sometimes its really really easy. For this equation it would be very easy, but for sin(x) * dy/dx + y = 1 it would be more difficult. So difficutly just depends on the particular equation.

 

[Integral]

For the most part normal algebraic equations can be solved nearly mechanically, follow the rules, crank the handle of the machine and out pots a solution.

Such is not the case for differential equations, there are methods to use if certian conditions are met, and there are some which simply cannot be solved analytically.

The difficuty comes in learning when to apply which methods.

The key to solving most differential equations is knowing the solution befor you start. When you solve an algebraic equation the solution is a simple variable, the solution to a DE is a function, for many types of DE we can recognize the general function which solves the equation.

For example a DE of the form

X(t)"+ λX(t) = 0

has a general solution

of X(t)= ACos(λt)+ Bsin(λt) OR
X(t) = Ae^λt+ Be-^λt

Where A and B are constants.

A and B cannot be determined with the information I have provided, the complete statement of a DE includes either boundary condions, that is the value of the solution at some point (usually an end point) or an initial value (if the independent varialbe is time) which specifies the value at some time.

I really cannot present a course in DE, but perhaps you can see parts of the quest that lie ahead of you.

To really get an understanding of DE you need to understand functions, you must have a mastery of algebra and a good understanding of calculus, both differential and integral.

Good luck.

 

[MathematicalPhysicist]

and here:http://www.sosmath.com/diffeq/first/lineareq/example1/example.html you can find 3 examples of de.
(there is also an explanation how to solve them).

 

[NEOclassic]

another "making it easy" factor

Hello string,
When it comes to diff eqn there is a cool tool called an integrating factor: e^[f'(x)]which is as important to calculus as L'Hospital's (that's how Google spells it)Rule is to topology or Avagadro's Number is to Chemistry. Properly used the factor works magically on EQs that appear to be insoluable.
The most important rule in EQ that a beginner needs to respect is "separation of variables: E.g., dy/dx = yx^2 multiply thru by "ydx" gives ydy =[x^2]dx Both sides are now integratable and are equal [excepting some arbitrary integrating constant]. Cheers Jim

 

[Brad_Ad23]

This prooves to be a useful site:

http://www.physics.ohio-state.edu/~physedu/mapletutorial/tutorials/diff_eqs/intro.html