- #1
physicskid
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Can anyone please tell me how differential equation works? I already understand derivative and integral work except this...
How does differential equation work?
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SUMMARY
Differential equations are mathematical equations that involve derivatives of functions, essential for solving various physics and chemistry problems. They can be categorized into ordinary differential equations (ODEs) and partial differential equations (PDEs), with ODEs typically expressed in the form \(\dot{y} = f(y, x)\). Solutions to these equations often depend on boundary conditions, and many differential equations lack analytic solutions. The existence of solutions can be determined using the Cauchy-Lipschitz theorem for ODEs and variational formulations for elliptic PDEs.
PREREQUISITES- Understanding of calculus, including derivatives and integrals.
- Familiarity with ordinary differential equations (ODEs) and partial differential equations (PDEs).
- Knowledge of boundary conditions and their role in defining solutions.
- Basic understanding of mathematical theorems such as Cauchy-Lipschitz theorem.
- Study the Cauchy-Lipschitz theorem and its applications in proving the existence of solutions for ODEs.
- Explore variational formulations and the Lax-Milgram theorem for solving elliptic partial differential equations.
- Learn about numerical methods for approximating solutions to differential equations that lack analytic solutions.
- Investigate the differences between ordinary and partial differential equations, focusing on their unique characteristics and solution methods.
Students and professionals in mathematics, physics, and engineering who are seeking to deepen their understanding of differential equations and their applications in real-world problems.
- #2
Feynman
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we use the didderential eq to solve a lot of physics or chemistry problems
- #3
Feynman
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we use the differential eq to solve a lot of physics or chemistry problems- #4
HallsofIvy
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If you have done calculus then you have already solve some "differential equations"!
For example, a typical calculus problem is "find the general anti-derivative of f(x)" (where f(x) is some given function). That is exactly the same as saying "find all functions, y(x), that satisfy y'(x)= f(x)" and that is a differential equation!
Of course, that's a relatively easy, one-step, problem- you just integrate. It's a little like solving a very simple numerical equation: 3x= 4. You just divide both sides by 3. Harder differential equations are like more complicated numerical equations, say 3x+ 7= 15 or x2+ 3x2+ 7x+ 5= 0. Such differential equation would tell you some property the function y and its derivatives satisfy: for example,
y" - 3y' + y2= 3x2 is a (very hard!) differential equation.
If you are asking for general ways to solve differential equations, I will quote Fermat: "there is insufficient room in this border..."! I myself took 2 courses in differential equations as an undergraduate and two more as a graduate student and I still have to muddle through sometimes (and the great majority of differential equations have NO analytic solution).
For example, a typical calculus problem is "find the general anti-derivative of f(x)" (where f(x) is some given function). That is exactly the same as saying "find all functions, y(x), that satisfy y'(x)= f(x)" and that is a differential equation!
Of course, that's a relatively easy, one-step, problem- you just integrate. It's a little like solving a very simple numerical equation: 3x= 4. You just divide both sides by 3. Harder differential equations are like more complicated numerical equations, say 3x+ 7= 15 or x2+ 3x2+ 7x+ 5= 0. Such differential equation would tell you some property the function y and its derivatives satisfy: for example,
y" - 3y' + y2= 3x2 is a (very hard!) differential equation.
If you are asking for general ways to solve differential equations, I will quote Fermat: "there is insufficient room in this border..."! I myself took 2 courses in differential equations as an undergraduate and two more as a graduate student and I still have to muddle through sometimes (and the great majority of differential equations have NO analytic solution).
- #5
Galileo
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To put it simply: A differential equation is an equation that contains derivatives of a function. The unknown is the function and a solution to the equation is a function that obeys the differential equation.
To solve a differential equation generally means to find all solutions to the equation.
Typically, your function obeys certain boundary conditions which uniquely define a solution.
All fundamental laws of physics are differential equations. A few examples:
F(x,t)=m\frac{d^2x}{dt^2}
\nabla \cdot \vec E(\vec r) = \frac{\rho(\vec r)}{\epsilon_0}
i\hbar \frac{d\psi(x,t)}{dt}=-\frac{\hbar^2}{2m}\frac{d^2\psi(x,t)}{dx^2}+V(x,t)\psi(x,t)
To solve a differential equation generally means to find all solutions to the equation.
Typically, your function obeys certain boundary conditions which uniquely define a solution.
All fundamental laws of physics are differential equations. A few examples:
F(x,t)=m\frac{d^2x}{dt^2}
\nabla \cdot \vec E(\vec r) = \frac{\rho(\vec r)}{\epsilon_0}
i\hbar \frac{d\psi(x,t)}{dt}=-\frac{\hbar^2}{2m}\frac{d^2\psi(x,t)}{dx^2}+V(x,t)\psi(x,t)
- #6
arildno
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If you want to understand the difference between a differential equation, and the more ordinary equations you've dealt with, consider the type of unknowns you're supposed to find:
a)In your ordinary equations, you're supposed to find some NUMBER "x" which fits into your equation.
Slightly more difficult, you're supposed to find NUMBERS "x and y" which fit simultaneously into the equation(s) you've been given.
In all such cases, you're supposed to find a set of numbers which fit your equations exactly.
b)FUNCTIONAL EQUATIONS on the other hand, are equations where your unknowns are functions, rather than single numbers.
There are lots of different functional equations; DIFFERENTIAL equations are the most "common" variety of functional equations.
As the other posters have said, what characterises a differential equation is that the derivatives of the function you seek is included in the functional equation you're trying to solve.
If both integrals and derivatives of your unkown function are included in your equation, you'll typically call it an integro-differential equation.
The other posters have given great examples of differential equations, so I'll stop here..
a)In your ordinary equations, you're supposed to find some NUMBER "x" which fits into your equation.
Slightly more difficult, you're supposed to find NUMBERS "x and y" which fit simultaneously into the equation(s) you've been given.
In all such cases, you're supposed to find a set of numbers which fit your equations exactly.
b)FUNCTIONAL EQUATIONS on the other hand, are equations where your unknowns are functions, rather than single numbers.
There are lots of different functional equations; DIFFERENTIAL equations are the most "common" variety of functional equations.
As the other posters have said, what characterises a differential equation is that the derivatives of the function you seek is included in the functional equation you're trying to solve.
If both integrals and derivatives of your unkown function are included in your equation, you'll typically call it an integro-differential equation.
The other posters have given great examples of differential equations, so I'll stop here..
Last edited:
- #7
physicskid
- 75
- 0
Thanks for all your responses! I now know the definition andthe examples of a differential equation. As for the rest of it, I'll read more about it.
- #8
Ethereal
When does a differential equation have an analytic solution? Could you tell simply by being unable to solve it, or is there a proof for each separate DE that it has no analytic solution?
- #9
Feynman
- 152
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What do you mean Ethereal ?
are you ask about the existence of the solution?
are you ask about the existence of the solution?
- #10
Ethereal
Yes, how do we know if a solution exists?Feynman said:
- #11
Feynman
- 152
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OHHHHHHHHHHHHHHH , it is a big problem
are u talking about differential or partial differential eq?
are u talking about differential or partial differential eq?
- #12
Ethereal
How about both?Feynman said:
- #13
Feynman
- 152
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their is a long theory about the existence of diff eq, and another one for partial diff eq
What is your target?
What is your target?
- #14
Feynman
- 152
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for Differential eq , all diff eq has the folowing form \dot{y}=f(y,x}
Where F is a differentiable fct,
to proove that this eq has a solution we use the theorem of Cauchy Lipscitz
that : f is a lip****zienne function
for partial differential,their is several problem , if the equation is elliptic we use the variational formulatrion then the Lax-Milgram theorem,
if it is on hyperpolic type and the parabolic we use the spectral theory
Where F is a differentiable fct,
to proove that this eq has a solution we use the theorem of Cauchy Lipscitz
that : f is a lip****zienne function
for partial differential,their is several problem , if the equation is elliptic we use the variational formulatrion then the Lax-Milgram theorem,
if it is on hyperpolic type and the parabolic we use the spectral theory
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