How to Convert a 2nd Order ODE System to 1st Order with Consistent Units?

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In summary, the conversation discusses the need to convert all measurements to the same unit in order to re-write a system of 1st order ODEs. This will make solving the ODEs easier.
  • #1
Dustinsfl
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\begin{alignat*}{3}
m\ddot{x} & = & -c(y)\sqrt{x^2+y^2}x\\
m\ddot{y} & = & -mg - c(y)\sqrt{x^2 + y^2}y
\end{alignat*}
where $c(y) = 0.25\text{N}\cdot\text{s}^2/\text{m}^4\cdot (15\text{cm})^2\exp(-y/(10000\text{m}))$
In order to re-write this as a system of 1st order ODEs, do I have to put everything in the right dimensions like all meters or centimeter measurements first?
 
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  • #2
Yes, it is important to make sure that all measurements are expressed in the same unit. This makes it easier to solve the system of ODEs. For example, if you want to convert from cm to m, you can multiply the measurements by $10^{-2}$.
 

FAQ: How to Convert a 2nd Order ODE System to 1st Order with Consistent Units?

What is the difference between 2nd order and 1st order?

The difference between 2nd order and 1st order is the number of variables in the equation. In a 2nd order equation, there are two variables (x² and x), while in a 1st order equation, there is only one variable (x).

How do you convert a 2nd order equation to a 1st order equation?

To convert a 2nd order equation to a 1st order equation, you can use the substitution method or the elimination method. The substitution method involves replacing one variable with another, while the elimination method involves cancelling out one variable by adding or subtracting equations.

What is the significance of 2nd order to 1st order in physics?

In physics, 2nd order to 1st order equations often represent the motion of objects. 2nd order equations are used to describe acceleration, while 1st order equations are used to describe velocity. By converting a 2nd order equation to a 1st order equation, we can better understand the motion of an object and make predictions about its future behavior.

Can a 2nd order equation have more than two variables?

Yes, a 2nd order equation can have more than two variables. This type of equation is known as a multivariate 2nd order equation. However, for simplicity, most 2nd order equations used in mathematics and physics have only two variables.

What are some real-life applications of 2nd order to 1st order conversion?

One real-life application of converting a 2nd order equation to a 1st order equation is in predicting the trajectory of a projectile. By converting the equation of motion from 2nd order to 1st order, we can determine the initial velocity and angle needed for a projectile to hit a specific target. This conversion is also used in solving problems related to electrical circuits and chemical reactions.

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