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So I've just begun a course on differential equations, I understand that a solution comes in the form of a function that satisfies the equation, however, I am still confused on what we use the solution to a differential equation for. I asked my teacher and he stated that the solution allows us to make future predictions on the thing we are trying to model, but I'm still confused on how a solution lets us make a future prediction of whatever we are trying to model?.

would a particular scenario of this be predicting weather forecasts??

MathLearner
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    The laws of physics come in the form of differential equations, for example Newton's second law is a second order ODE for the position of an object. If you could solve for the position function due to an external force then you'd know the future path of the object. The well known parabolic path of objects close to earth is one such result. – Triatticus Jan 20 '17 at 06:06
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    What are solutions to anything for then? – JobHunter69 Jan 20 '17 at 06:28
  • Multiple physical systems. You'll now know why a spring system can be modeled with trigonometry. You'll know why populations can be modeled with exponential functions. You can now include air resistance in your calculations. You can now make more sense of systems modeled with time. Once you learn the applications of ODEs, you'll wonder how you did applied math without them (People solved the ODEs and then spoonfed you the final equations, that's how) – Kaynex Jan 20 '17 at 06:32
  • Weather forecasts are done through what we currently understand about the Navier-Stokes equation, which is a partial differential equation. Insight into this equation is currently a $1,000,000 question. – Kaynex Jan 20 '17 at 06:34
  • ODEs come into picture when you have good insight into the dynamics of a system, then attempt to derive a static description of it. For a simple non-physics example, lookup pursuit curves. For a light-hearted application thereof, see my answer to Escaping from a circle of fat lions. – dxiv Jan 20 '17 at 07:55
  • Any response to the answers, MathLearner? – Gerry Myerson Jan 22 '17 at 00:35

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If you have an unknown number $x$, but you have an equation it satisfies, say, $x^2-5x+6=0$, then you can use the equation to work out what number $x$ is.

If you have an unknown function $f$, but you have a differential equation it satisfies, say, $f''-5f'+6f=0$, you can use the differential equation to work out what function $f$ is.

And once you know what function $f$ is (that is, once you have a formula for $f$), you can calculate the value of $f(t)$ for any $t$ you like.

Gerry Myerson
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An common application to ODE's is in physics. For example, you can add drag force in a calculation, whereas Newton's laws of constant acceleration do not: $$s=ut+\frac{1}{2}at \tag{1}$$ Without drag, the differential equation for earth's gravity which would satisfy this is: $$F=-F_g$$ $$m\frac{d^2 s}{dt^2}=-mg$$ $$\frac{d^2 s}{dt^2}=-g \tag{2}$$ This differential equation can be solved using various techniques, to derive equation $(1)$.

However, it gets more interesting when you do include drag force (air resistance) in your calculation.

The total force $F$ is given by:

$$F=-F_g+F_d$$

Where $F_g$ is the force due to gravity and $F_d$ is the force due to drag. We can apply our knowledge of forces to form the following differential equation:

$$m\frac{d^2 s}{dt^2}=-mg+k\left(\frac{ds}{dt}\right)^2 \tag{3}$$

Solving the above differential equation and applying appropriate initial conditions should give you the position of the falling object at any given point in time under air resistance by means of finding the unknown function $s(t)$.

These differential equations can be made to predict the weather such as temperature, by evaluating the function for $T(t)$ on a differential equation, where $T$ is the temperature and $t$ is the time.

projectilemotion
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