lightlightsup said:Homework Statement:: ##e^{-0.6x}\sin{(5x)}-0.1=0##
Homework Equations:: https://www.desmos.com/calculator/ycgu3jt9qd
I have posted my graphical solution to this problem.
But, how do I solve this numerically/mathematically without graphing it?
It seems to me that the OP is trying to solve the equation shown above.alan2 said:What is your question?
I would still like to know the statement of the problem.lightlightsup said:You've both answered my question. Thank You.
Guess you spoke too soon, see his response above. He obviously wasn’t being asked to solve for x, it can’t be done analytically, don’t know why you would think that was the question. Of course it has everything to do with an ODE, it’s the solution to a second order equation with damping.HallsofIvy said:The "statement of the problem"
would be "Solve the equation [itex]e^{-0.6x}sin(5x)- 0.1= 0[/itex] for x". While that was not said in so many words, when a post shows an equation in "x" and asks "how do you solve this" that's pretty much implied! In your first response you seem to have the idea that this question was about differential equations. I have no idea where you got that.
The envelope will decay to 10% of its max when the exponential term is equal to 0.1. Then just take the natural log and solve for t.lightlightsup said:The ##x## above is ##t(time)##.
If you guys are wondering where this equation came from: simple damped spring oscillations, where one is solving for when a certain amplitude will be reached (or a certain percentage of a max amplitude).
The reality is that this amplitude may never actually be reached at a certain time.
But, the max amplitude envelope will be reached at that time ##t##.
So, if a damped oscillation is given by: ##x(t)=0.16e^{-0.6t}\sin{(5t)}##.
When do you reach 10% of the max amplitude (0.16 meters)?
I just wanted to confirm that I haven't yet learned the math required to solve these equations non-graphically.
Damped Oscillations Link
Here is is response from post #4:alan2 said:Guess you spoke too soon, see his response above.
lightlightsup said:You've both answered my question. Thank You.
He obviously was asking about how to solve the equation he wrote, either analytically or numerically. Here is what he wrote in post #1.alan2 said:He obviously wasn’t being asked to solve for x, it can’t be done analytically, don’t know why you would think that was the question.
I made several suggestions of techniques that could be used for solving the equation numerically.lightlightsup said:Homework Statement:: ##e^{-0.6x}\sin{(5x)}-0.1=0##
But, how do I solve this numerically/mathematically without graphing it?
But that's not the question he asked.alan2 said:Of course it has everything to do with an ODE, it’s the solution to a second order equation with damping.
Damping in an equation refers to the gradual decrease of a variable over time. It is often represented by the coefficient "d" in an exponential-decay equation, and it determines how quickly the variable decreases.
To solve for damping in an equation, you will need to know the initial value of the variable, the rate at which it decreases (usually represented by "r"), and the time elapsed. You can then use the formula d = ln(initial value/final value)/time to calculate the damping coefficient.
An exponential-decay equation is a mathematical expression that models the decrease of a variable over time. It follows the form y = Ae^-rt, where y is the final value, A is the initial value, r is the rate of decay, and t is time.
To use an exponential-decay equation to solve a real-world problem, you will first need to identify the variables involved (such as the initial and final values, rate of decay, and time). You can then plug these values into the equation and solve for the unknown variable.
Solving exponential-decay equations is useful in many fields of science and engineering, such as physics, chemistry, and biology. It can be used to model radioactive decay, population growth and decline, and the dissipation of energy in mechanical systems.