Solving a complex equation (damping/exponential-decay) like this....?

[lightlightsup]

Homework Statement

$e^{-0.6x}\sin{(5x)}-0.1=0$

Relevant Equations

https://www.desmos.com/calculator/ycgu3jt9qd

$e^{-0.6x}\sin{(5x)}-0.1=0$
I have posted my graphical solution to this problem.
But, how do I solve this numerically/mathematically without graphing it?

 

[alan2]

What is your question? What you have is the solution to a second order ordinary differential equation with initial conditions. What are you trying to solve?

 

[Mark44]

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?

What is your question?

It seems to me that the OP is trying to solve the equation shown above.

Regarding find a solution numerically, most textbooks on numerical methods list several techniques for find zeroes of equations, such as bisection, regula falsi (false position), Newton-Raphson, and others. See https://en.wikibooks.org/wiki/Numerical_Methods/Equation_Solving for examples of these techniques.

As far as analytic solutions go, there is the Lambert w function, but your equation is complicated enough that that approach might not be useful.

 

[lightlightsup]

You've both answered my question. Thank You.

 

[alan2]

You've both answered my question. Thank You.

I would still like to know the statement of the problem.

 

[HallsofIvy]

The "statement of the problem"
would be "Solve the equation e^{-0.6x}sin(5x)- 0.1= 0 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.

 

[lightlightsup]

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

 

[alan2]

The "statement of the problem"
would be "Solve the equation e^{-0.6x}sin(5x)- 0.1= 0 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.

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.

 

[alan2]

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

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.

 

[Mark44]

Guess you spoke too soon, see his response above.

Here is is response from post #4:

You've both answered my question. Thank You.

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.

He obviously was asking about how to solve the equation he wrote, either analytically or numerically. Here is what he wrote in post #1.

Homework Statement:: $e^{-0.6x}\sin{(5x)}-0.1=0$
But, how do I solve this numerically/mathematically without graphing it?

I made several suggestions of techniques that could be used for solving the equation numerically.

Of course it has everything to do with an ODE, it’s the solution to a second order equation with damping.

But that's not the question he asked.