# What is a differential equation and what use do they have?

I thought I've seen one, but I do not understand exactly how they negate.

For example, if you have $$d/dx^2 + 1$$, does the dx cancel?

arildno
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What do you know about derivatives?

arildno said:
What do you know about derivatives?

Quite a bit, I understand that deriviatives are an infinitely small section of a curve, making up the entire curve with an infinite amount of infinltey small sections.

Derivatives give the rate of change of one variable with respect to another, hence they can be used to find the slope of curves. Differential equations are equations involving derivatives and have lots of applications in physics. Newton's 2nd law can be expressed as a second order differential equation: F = m (d^2x/dt^2). In electric circuits, we can use differential equations to find out how current changes as a function of time, for example in a circuit involving an inductor. There's also the Schrodinger equation, in quantum mechanics that can tell us about energy for a quantum system. The diffusion and wave equations are other examples.

Integral
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More generally, Differential equations are basis for every meaningful physical theory in existence. They are the starting point for the math. DEqs are the mathematical language to express how things change. All of our physical observations are in the terms of how things change. If you translate the observed changes in the physical world carefully into mathematics you have a differential equation. Solve the differential equations and you have a tool to make a physical prediction.

dextercioby
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Nylex said:
Derivatives give the rate of change of one variable with respect to another, hence they can be used to find the slope of curves. Differential equations are equations involving derivatives and have lots of applications in physics. Newton's 2nd law can be expressed as a second order differential equation: F = m (d^2x/dt^2). In electric circuits, we can use differential equations to find out how current changes as a function of time, for example in a circuit involving an inductor. There's also the Schrodinger equation, in quantum mechanics that can tell us about energy for a quantum system. The diffusion and wave equations are other examples.

I'm sorry,i'd have to contradict you here.The mathematical expression for Newton's second law (linear motion) is actually:
$$\frac{d\vec{p}}{dt} = \sum_{i} \vec{F}_{i}$$
,where 'i' is an index which takes values in a subset of N.
Try to figure out why i've written it in such form.

dextercioby said:
I'm sorry,i'd have to contradict you here.The mathematical expression for Newton's second law (linear motion) is actually:
$$\frac{d\vec{p}}{dt} = \sum_{i} \vec{F}_{i}$$
,where 'i' is an index which takes values in a subset of N.
Try to figure out why i've written it in such form.

but how can you express the force on an object. It can be expressed as

d^2 (x) / d (t^2).

- harsh

Tide
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Harsh,

That's true only when the mass is constant in which case

$$\frac {d\vec p}{dt} = \frac {d m \vec v}{dt} = m \frac {d \vec v}{dt} = m \frac {d^2\vec x}{dt^2}$$

Tide, harsh, and dextercioby are all basically saying the same thing, but let me provide some elucidation. As a historical note, Newton wrote his second law originally in terms of change-of-momentum. That is, the net force acting on an object causes a change in momentum as follows (where the net force and momentum are both vector quantities):

$$\vec{F}_{net} = \frac{d}{dt}(\vec{p}) = \frac{d}{dt}(m\vec{v})$$ whereby $$\vec{F}_{net} = \sum_i{\vec{F}_i}.$$

For the non-relativistic case ($$v << c$$), we can add the approximation that the mass is a constant and treat it as such. Of course, for relativistic speeds (usually $$v \ge 0.1c$$), we need to invoke the special relativity mass correction factor of $$m = m_0\frac{1}{\sqrt{1-(v/c)^2}}.$$

How this all applies to differential equations is that the physical phenomenon of nature are modeled by differential equations. Newton's second law above gives a great example. Take the motion of a mass resting on a flat frictionless tabletop that is displaced from its equilibrium position by some deflection $$x$$. Now, the primary force we're concerned with is the restoring force of the spring, given by $$F = -kx$$ where I drop the vector symbols because we only have 1-dimensional freedom of movement and the direction is indicated by sign. Now, by Newton's second law, this gives us the following equation:

$$F_{net} = \frac{d}{dt}(mv) \implies -kx = mx'' \implies x'' + \frac{k}{m}x = 0.$$

Voila. This is a linear second order differential equation with constant coefficients that models the motion of the mass. Two solutions of this differential equation are:

$$x(t) = \sin{\left(t\sqrt{k/m}\right)}$$ and $$x(t) = \cos{\left(t\sqrt{k/m}\right)}$$

and by an elementary theory from differential equations, we know that all solutions of the system can be written in the form of

$$x(t) = C_1\sin{\left(t\sqrt{k/m}\right)} + C_2\cos{\left(t\sqrt{k/m}\right)}$$

where $$C_1$$ and $$C_2$$ are constants determined by the initial conditions of the system. A knowledge of physics and differential equations combined is a powerful tool for producing accurate models of the physical universe.

As for your comment about derivatives and infinities, this is bordering into the philosophy of mathematics, which has a very long history. Debates around infinity and infinitesimals still go on today despite our everyday application of them. Strictly speaking, mathematicians use the term "differential" to describe small changes in one quantity (e.g. $$dx$$) whereas they use the term "derivative" to refer to the operation of differentiation.

Cheers.
---
Mike Fairchild
http://www.mikef.org/
"Euclid alone has looked on beauty bare."
--Edna St. Vincent Mallay

Tide
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Mike

It's possible for the mass to be a variable even for strictly classical physics problems such as in the case of a rocket expelling mass as it accelerates.

dextercioby
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mjfairch said:
Tide, harsh, and dextercioby are all basically saying the same thing, but let me provide some elucidation. As a historical note, Newton wrote his second law originally in terms of change-of-momentum. That is, the net force acting on an object causes a change in momentum as follows (where the net force and momentum are both vector quantities):

$$\vec{F}_{net} = \frac{d}{dt}(\vec{p}) = \frac{d}{dt}(m\vec{v})$$ whereby $$\vec{F}_{net} = \sum_i{\vec{F}_i}.$$

For the non-relativistic case ($$v << c$$), we can add the approximation that the mass is a constant and treat it as such. Of course, for relativistic speeds (usually $$v \ge 0.1c$$), we need to invoke the special relativity mass correction factor of $$m = m_0\frac{1}{\sqrt{1-(v/c)^2}}.$$

How this all applies to differential equations is that the physical phenomenon of nature are modeled by differential equations. Newton's second law above gives a great example. Take the motion of a mass resting on a flat frictionless tabletop that is displaced from its equilibrium position by some deflection $$x$$. Now, the primary force we're concerned with is the restoring force of the spring, given by $$F = -kx$$ where I drop the vector symbols because we only have 1-dimensional freedom of movement and the direction is indicated by sign. Now, by Newton's second law, this gives us the following equation:

$$F_{net} = \frac{d}{dt}(mv) \implies -kx = mx'' \implies x'' + \frac{k}{m}x = 0.$$

Voila. This is a linear second order differential equation with constant coefficients that models the motion of the mass. Two solutions of this differential equation are:

$$x(t) = \sin{\left(t\sqrt{k/m}\right)}$$ and $$x(t) = \cos{\left(t\sqrt{k/m}\right)}$$

and by an elementary theory from differential equations, we know that all solutions of the system can be written in the form of

$$x(t) = C_1\sin{\left(t\sqrt{k/m}\right)} + C_2\cos{\left(t\sqrt{k/m}\right)}$$

where $$C_1$$ and $$C_2$$ are constants determined by the initial conditions of the system. A knowledge of physics and differential equations combined is a powerful tool for producing accurate models of the physical universe.

As for your comment about derivatives and infinities, this is bordering into the philosophy of mathematics, which has a very long history. Debates around infinity and infinitesimals still go on today despite our everyday application of them. Strictly speaking, mathematicians use the term "differential" to describe small changes in one quantity (e.g. $$dx$$) whereas they use the term "derivative" to refer to the operation of differentiation.

Cheers.

I'm sorry but i wasn't referring at all to the relativistic case (and maybe neither Arildno),but to a very classical case,analyzed at the end of the 19-th century (round 1892,if i'm not mistaking) of the Newton's equations.

Does the name Tsiolkowskiy tell u something????????????Guess not,else u would have realised i was referring to his famous equation which describes the movement of a rocket.Yes,you saw well,the first rockets ever launched were the famous V1 and V2-s of the German Army (Wehrmacht) in 1941 in the Battle of England (actually they launched a much more massive attack later,towards the end of thewar).Anyway,the theory of rocket trajectories had been done many years before (round 50) by the russian Tsiolkowskiy.He bassically assumed that Newton equation would be written for a classical body with varying mass,viz.

$$\frac{dm(t)}{dt}\vec{v}+m(t)\frac{d\vec{v}}{{dt} =m{t}\vec{g}+\vec{F}_{friction forces}$$

Daniel.

PS.I guess he assumed the friction forces Stokes type.

EDIT:Arildno was referring at the same thing. :tongue2:

Last edited:
Sure, all of this is true. Variable mass systems are common as well and serve as another example to demonstrate the usefulness of differential equations in modeling physical systems. Anyway, my point to the original post of this thread is that they're indispensible in the study of physical phenomenon, and I believe the example provided by others as well as the simple example I gave should give a taste of their form and application.

Cheers.
---
Mike Fairchild
http://www.mikef.org/
"Euclid alone has looked on beauty bare."
--Edna St. Vincent Mallay

saltydog
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Nicely put!

Integral said:
More generally, Differential equations are basis for every meaningful physical theory in existence. They are the starting point for the math. DEqs are the mathematical language to express how things change. All of our physical observations are in the terms of how things change. If you translate the observed changes in the physical world carefully into mathematics you have a differential equation. Solve the differential equations and you have a tool to make a physical prediction.

Go get um' !
SD

mathwonk
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2020 Award
newtons law F = mA says that forces (F) produces changes in velocity(A). Thus observing forces at work can allow you to predict the future position of something. Being able to predict the future is useful.

I have a small question. Is 'Differential Equation' part of 'Calculus'?

I think that normal equations (or formula) gives us the average velocity,accelerations and what else.As we know from limits,derivatives is about rate of change.It means we are looking for instantaneous velocity or acceleration.We know that all quantities doesn't contain the derivatives.For example : F=ma a is second derivatives of x.but mass is not derivative of anything.it's not changing.Also we know from derivatives that we need a boundary conditions.For f=ma --> a is depend to t.Acceleration is changing with time.(t is boundary condition)

As a result,I think this is a reson of Differential Equations.