How to code second order differential equation with input variable t?

Hello, how do I implement the following second order differential equation with input variable t (0 to 300) and output the corresponding displacement h? Thank you.

1369.8*h + 2566.7*hdot + 2.785e7*hdotdot - 2091.2*exp(-0.59452*t)*(exp(0.59452*t) + 1.0) = 0

Answer 1

[Mark Rzewnicki]

Looks like we have a 2nd-order differential equation of function . To clean the problem up, let's define some constants:

, , , ,

Then we have the following expression:

Now get by itself so we can use a numerical method to integrate the system:

Suppose you define . Then the system is determined for when you specify the initial conditions: .

Notice that .

Let a very small change in t be denoted . Then you could approximate the value of using the relation .

So the process looks like this:

1. Start at and specify the initial conditions and . This allows you to calculate .

2. Use and to determine and h after a tiny step . Now you can calculate .

3. Use and to determine and h after another tiny step . Update the value of .

4. Repeat until .

This process is called "rectangular integration" and it allows you to numerically solve DEs quickly. If you want higher accuracy, you can use a Runge-Kutta method. These methods follow a very similar procedure but reduce the estimation error at every step. A popular one is a fourth-order method commonly called "RK4."

EDIT: I played around simplifying the original equations. Notice that .

[Saruultugs Batzorig]

Thank you.