SimReal / Python / Matlab / JSXGraph: Programming - Exercise - Differential Equation

SimReal - Python - JSXGraph

001. ODE - 001

Solve the following differential equation:

dy/dt = -k*y

where
y = y(t)
k = 0.2
y(0) = 3

SimReal - Python - JSXGraph

002. ODE - 002

Solve the following differential equation:

dy/dt = -k*y

where
y = y(t)
k = 0.1, 0.2, 0.3
y(0) = 3

SimReal - Python - Matlab

003. ODE - 003

Sove the following differential equation:

5dy/dt = - y/t) + u(t)

where
y = y(t)
y(0) = 3
u er step funksjon fra 0 til 2 ved t = 10

SimReal - Python - Matlab

004. ODE - 004

Find the solution of the following differential equations:

dx/dt = 3e^-t
dy/dt = 3 - y(t)

where
x(0) = 0 and y(0) = 0

SimReal - Python - Matlab

005. ODE - 005

Find the solution of the following coupled ordinary differential equations including a step function:

2*dx/dt = -x(t) + u(t)
5*dy/dt = -y(t) + x(t>
where
x(0) = 0 and y(0) = 0
u = u(t) = 2*S(t-5) where S(t-a) is the step function that changes from 0 to 1 at t = a

SimReal - Python - JSXGraph

006. Second Order Differential Equation

Find the solution of the following second order differential equation:

y'' + 2y' + y = x

with initial conditions:
y(x₀) = y(0) = 5
y'(x₀) = y'(0) = 3

SimReal - Python - Matlab

007. ODE - Boat

A boy is standing in the origin A. He has a boat originally in the position B.
The boy is moving along the x-axis (the horisontal axis) keeping the string taut (blue color).

Simulate this experiment by the help of small object (boat), pen and paper on a horizontal area.
Mark 10 points on the path of the boat.
Especially notice the point when the boat is 6 m from the horizonal line from A.
Notice how far from the verical line AB the boat is now and at what position E the boy is right now.

Describe by a mathematical model (differential equation) the path (red color) that the boat will follow.

Use the SimReal-application above to:
- Input the mathematical model
- Find the numerical solution of the path
- Draw the graph
- Simulate the movement of the boat
- Find out how far from the vertical line the boat is when the boat is located 6 m from the horisontal line AE
- What is the position of the point E (boy with the rope) now?
- Compare the result from the experiment and from the mathematical model