Solving Differential Equations: Understanding Variable Changes

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In summary, the conversation discusses the concept of differentiating equations, specifically taking the derivative of both sides of an equation. It is noted that for equations with multiple variables, the derivative will have multiple terms. The concept of changing the integrating variable is also mentioned, and it is confirmed that this can be done in any way as long as it follows the rules of differentiation. The example of the "rocket equation" is used to illustrate this concept. The use of the product rule in differentiation is also mentioned.
  • #1
bassplayer142
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A little thing that may be stupid but I am confused about it. Say we take any equation like f=ma
If we take the derivative of both sides then we could either have

df=m da
or
df =a dm

Are both of these valid computations. If I am looking to change the integrating variable can I use this any way I want? And would this work with any equation relating 3 or more variables?

thanks in advanced
 
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  • #2
You are basically correct: the "rocket equation" comes from F = dp/dt = d(mv)dt = m dv/dt + v dm/dt.

For what you wrote, f = ma, df = m da/dt + a dm/dt.
 
  • #3
If f=abc, df=ab dc+ ac db + bc da, and so on for an number of variables.
You just differentiate one at a time.
Of course if any factor is a constant, then its differential is zero.
 
  • #4
Andy Resnick said:
You are basically correct: the "rocket equation" comes from F = dp/dt = d(mv)dt = m dv/dt + v dm/dt.
The "rocket equation" is a bit different, because the exit velocity of the gas enters instead of just v in the dm/dt term.
 
  • #5
thanks this clears some up. But as I said before could I take df=mda to subsitute df with da to integrate with respect to a, and in the same problem could I take df=adm to integrate with respect to m.

so you could take s=rTheta and make it ds=rdtheta?

I'm just seeing how flexible I can be when substituting vaiables to integrate or differentiate with.

Edit, I just realized that what you did there was the product rule which makes sense. Is what I just said above wrong then?>
 
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  • #6
A derivative of all product of N variables will have N terms, each term being differentilated once. So d(r theta)=r dtheta+theta dr.
 

FAQ: Solving Differential Equations: Understanding Variable Changes

1. What is a simple differential question?

A simple differential question is a type of question that involves finding the derivative or rate of change of a function. It typically asks for the slope of a tangent line at a given point or the instantaneous rate of change at a specific moment.

2. How do I solve a simple differential question?

To solve a simple differential question, you first need to identify the function and the variable you are taking the derivative with respect to. Then, use the appropriate rules and formulas to find the derivative. Finally, substitute the given values into the derivative to find the solution.

3. What is the difference between a simple differential question and an integral?

A simple differential question involves finding the derivative, which is the rate of change of a function. An integral, on the other hand, involves finding the area under a curve. In other words, a derivative tells us the slope of a function, while an integral tells us the accumulation of a function.

4. Can simple differential questions be applied in real-life situations?

Yes, simple differential questions can be applied in various real-life situations, such as in physics, engineering, and economics. For example, the rate of change of position with respect to time is used to calculate velocity and acceleration in physics.

5. What are some common applications of simple differential questions?

Some common applications of simple differential questions include optimization problems, related rates, and curve sketching. These types of questions can help us analyze and understand real-life scenarios and make predictions based on the rate of change of a specific variable.

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