Hi! Can someone explain about Differential Equations?

[Sandra Jane]

Homework Statement

What is exactly Differential Equations?

Relevant Equations

Please use general equation examples so I can understand

So I'm new to engineering and have studied some of the calculus but until now, I still have a hard time to understand what is exactly Differential Equations, what is it for and how can I use it in the future classes as an Engineering Physics student

 

[PeroK]

This site is a good reference for all things calculus:

https://tutorial.math.lamar.edu/classes/de/de.aspx

 

[256bits]

Hi there Sandra Jane.

Have you tried some sites in the internet such as an introduction to differential equations with some neat real world applications.
https://www.mathsisfun.com/calculus/differential-equations.html

and something more in depth:
https://www.geeksforgeeks.org/maths/differential-equations/

 

[berkeman]

Welcome to PF.

Differential Equations, what is it for and how can I use it in the future classes as an Engineering Physics student

You will use Differential Equations a lot in Engineering Physics. For example, you will analyze electronic circuits that have inductors and capacitors in them using Differential Equations that describe the voltages and currents and how they vary with time.

https://en.wikipedia.org/wiki/Inductor

https://en.wikipedia.org/wiki/Capacitor

 

[FactChecker]

It's an equation where the change of a variable is involved. That is very common and there are thousands of examples. Here are a couple simple ones:
1) The population, $P$, of a country changes based on birth rate, $r_{babies}$, and death rate, $r_{deaths}$. So we get an equation like $dP/dt = r_{babies}*P -r_{deaths}*P$
2) The average air temperature, $T$, changes based on the amount of sunlight, $S$. So we get an equation like $dT/dt = \lambda S$.

 

[DaveE]

https://www.khanacademy.org/math/ap.../bc-7-1/v/differential-equation-introduction?

 

[PeroK]

It may be worth noting that Newton's second law is a differential equation, as it involves the acceleration of a particle ( which is second time derivative of displacement):
$$F = ma = m\frac{d^2x}{dt^2}$$In many cases, the force is a function of displacement. E.g. in a spring or simple pendulum we have $F =-kx$, where $k$ is a positive constant. This leads to the differential equation for simple harmonic motion:
$$\frac{d^2x}{dt^2} = -\frac k m x$$That might be the first differential equation that a lot of students encounter.

 

[sbrothy]

It may be worth noting that Newton's second law is a differential equation, as it involves the acceleration of a particle ( which is second time derivative of displacement):
$$F = ma = m\frac{d^2x}{dt^2}$$In many cases, the force is a function of displacement. E.g. in a spring or simple pendulum we have $F =-kx$, where $k$ is a positive constant. This leads to the differential equation for simple harmonic motion:
$$\frac{d^2x}{dt^2} = -\frac k m x$$That might be the first differential equation that a lot of students encounter.

I like Newton's law as an example of a simple differential equation. I find it easy to wrap my head around it. Later ones not so much, but it's a good start.

EDIT: And you say exactly that. Sorry. Didn't see your last sentence. So typical of me. I really have to work on that.

 

[sbrothy]

Since physics is fundamentally about change—motion, heat flow, electric fields, waves, quantum systems—differential equations become one of the main languages of physics and engineering.

[...]

Physics is mostly about change, so physics naturally leads to differential equations.

Indeed. And as long as they're ODEs I might have a chance. With PDEs I'm quickly unmasked as the autodidact hack I really am.

How Ordinary Differential Equations Differ from Partial Differential Equations

 

[Sandra Jane]

I like Newton's law as an example of a simple differential equation. I find it easy to wrap my head around it. Later ones not so much, but it's a good start.

EDIT: And you say exactly that. Sorry. Didn't see your last sentence. So typical of me. I really have to work on that.

[...]


Indeed. And as long as they're ODEs I might have a chance. With PDEs I'm quickly unmasked as the autodidact hack I really am.

How Ordinary Differential Equations Differ from Partial Differential Equations

Really appreciate the explanation. Thank you so much XD

 

[robphy]

Possibly enlightening:

​

Differential equations, a tourist's guide | DE1
3Blue1Brown

 

[mathwonk]

watching the movie Ivanhoe today I was interested in the height of the famous fall off the castle by the great stuntman Paddy Ryan, reputed to be 100 feet into a moat. So I timed it at about 2.5 seconds. Given Newton's law (which goes back to Galileo) the distance fallen in 2.5 seconds should be about 16(2.5)^2 =100 feet! I.e. if you know the second derivative of the distance formula to be the acceleration due to gravity = g ≈ 32 ft/sec^2, you can deduce that the distance formula is (1/2)gt^2 ≈ 16t^2, where t = time in seconds. So differential equations allow one to answer such questions.
[Perhaps I am confusing Newton's law with Galileo's simpler principle of constant gravitational acceleration, but I am not a physicist.]

 

[Kirti_Vardhan_1]

Yes.

The equation

$$
y = x - 2
$$

is an algebraic equation because it directly relates the variables $x$ and $y$ and contains no derivatives.

In contrast,

$$
\frac{dy}{dx} = x + 2
$$

is a differential equation because it contains the derivative $\frac{dy}{dx}$, which represents the rate of change of $y$ with respect to $x$.

A useful distinction is that an algebraic equation usually describes a specific relationship between variables, whereas a differential equation describes how a quantity changes and typically has a family of solutions.

For example, solving

$$
\frac{dy}{dx} = x + 2
$$

gives

$$
y = \int (x+2)\,dx
$$

$$
y = \frac{x^2}{2} + 2x + C
$$

where $C$ is an arbitrary constant.

This shows that a differential equation generally corresponds to infinitely many possible functions rather than a single curve. Each choice of $C$ produces a different solution curve, all of which satisfy the differential equation.

 

[PeroK]

Yes.

The equation

$$
y = x - 2
$$

is an algebraic equation because it directly relates the variables $x$ and $y$ and contains no derivatives.

In contrast,

$$
\frac{dy}{dx} = x + 2
$$

is a differential equation because it contains the derivative $\frac{dy}{dx}$, which represents the rate of change of $y$ with respect to $x$.

A useful distinction is that an algebraic equation usually describes a specific relationship between variables, whereas a differential equation describes how a quantity changes and typically has a family of solutions.

For example, solving

$$
\frac{dy}{dx} = x + 2
$$

gives

$$
y = \int (x+2)\,dx
$$

$$
y = \frac{x^2}{2} + 2x + C
$$

where $C$ is an arbitrary constant.

This shows that a differential equation generally corresponds to infinitely many possible functions rather than a single curve. Each choice of $C$ produces a different solution curve, all of which satisfy the differential equation.

Good answer, but you are a dollar short here and there.

 

[berkeman]

Good answer, but you are a dollar short here and there.

Fixed (Changed $ to double-# for inline LaTeX).

 

[Mark44]

Good answer, but you are a dollar short here and there.

And a day late?

 

[MidgetDwarf]

Even easier. Open up a book on differential equations. Its definition should be listed there.

 

[Matterwave]

watching the movie Ivanhoe today I was interested in the height of the famous fall off the castle by the great stuntman Paddy Ryan, reputed to be 100 feet into a moat. So I timed it at about 2.5 seconds. Given Newton's law (which goes back to Galileo) the distance fallen in 2.5 seconds should be about 16(2.5)^2 =100 feet! I.e. if you know the second derivative of the distance formula to be the acceleration due to gravity = g ≈ 32 ft/sec^2, you can deduce that the distance formula is (1/2)gt^2 ≈ 16t^2, where t = time in seconds. So differential equations allow one to answer such questions.
[Perhaps I am confusing Newton's law with Galileo's simpler principle of constant gravitational acceleration, but I am not a physicist.]

You already solved the differential equation here so it might be hard to tell what it was.

The differential equation originates from $F=ma$ and you assume a constant gravitational acceleration $g$ to get $a=g$. Then, you note $a=\frac{d^2 x}{dt^2}=x''(t)=g$. Since $g$ is constant you can integrate twice to get $x(t)=\frac{1}{2}gt^2+x'(0)t+x(0)$ you set the initial conditions to 0 and proceed from there.