Differential equation, recursive

Karl Karlsson

New member
We will look at a model with a number of driverless cars that use the same control system. Suppose that M cars drive one after another on a road. The position of car i at time t is called xi(t). The cars are arranged so that

The function g is just equal to g = 5 m/s for all t.

The function f is f(x)=x/3, for 0<x<75 and is for 75<x equal to 25.

Let M = 10

0 < t < 40s

Let the positions of the cars at time t = 0 be given by xi (0) = d · i, i = 1, ..., M

You are supposed to implement eulers method to solve this problem:
This problem is solved with paper and pencil. If the time step h is too large, the cars may "pass each other" in the Euler solution, ie. xi^n> x(i + 1)^n (this is not x times i + 1 nor is i+1 an input for the function, this only represents the (i+1):th x) , for some time step n. Determine an upper limit for time step h such that it is certain that this will not happen.

I am having problem with this as there are no similar problems in my book

Regards Karl

Karl Karlsson

New member
Does anybody have any idea?

Subhotosh Khan

Super Moderator
Staff member
We will look at a model with a number of driverless cars that use the same control system. Suppose that M cars drive one after another on a road. The position of car i at time t is called xi(t). The cars are arranged so that

View attachment 15074

View attachment 15076

View attachment 15075

The function g is just equal to g = 5 m/s for all t.

View attachment 15077

The function f is f(x)=x/3, for 0<x<75 and is for 75<x equal to 25.

Let M = 10

0 < t < 40s

Let the positions of the cars at time t = 0 be given by xi (0) = d · i, i = 1, ..., M

You are supposed to implement eulers method to solve this problem:
This problem is solved with paper and pencil. If the time step h is too large, the cars may "pass each other" in the Euler solution, ie. xi^n> x(i + 1)^n (this is not x times i + 1 nor is i+1 an input for the function, this only represents the (i+1):th x) , for some time step n. Determine an upper limit for time step h such that it is certain that this will not happen.

I am having problem with this as there are no similar problems in my book

Regards Karl
I am not sure I understand the problem!

On a different note -

You did not respond to our tutor's request in: