Approximating k with Shooting Method and Euler's Method

[androol]

dy/dt = ky, where k is a constant.
y|t=0 = 1; y|t=10 = 4

I need to approximate k using the shooting method and Euler's method... please help.. :)

wat's the procedure to approximate k using the shooting method

 

[HallsofIvy]

Do you know what "Euler's method" is?

Start with an easy estimate: take dt= 5 so we have only two steps
Initially, t= 0, y= 1, so dy/dt= k and dy= 5k

then t= 5, y= 1+ dy= 1+ 5k so dy/dt= k(1+ 5k)= k+ 5k² and so dy= 5(k+ 5k²)= 5k+ 25k².

Finally, then, x= 10, y= 1+ 5k+ dy= 1+ 5k+ 5k+ 25k²= 1+ 10k+ 26k²= That is, 25k²+ 10k- 3= 0. Solve that for k.

Now that you know, roughly, k, use, say, 10 steps from x= 0 to x= 10 with that k and see if you get 4 there. You won't of course, but that will tell you whether to make k smaller or larger. Do that repeatedly until you get a good method.

 

[tacman]

The shooting method is a numerical technique used to approximate the value of a constant in a differential equation. In this case, we are trying to approximate the value of k in the equation dy/dt = ky. The shooting method involves using an initial guess for the value of k and then solving the differential equation using that guess. The resulting solution is then compared to the desired boundary condition (in this case, y|t=10 = 4). If the solution does not match the boundary condition, the initial guess for k is adjusted and the process is repeated until a satisfactory approximation is found.

Euler's method is a numerical method for solving differential equations. It involves breaking the interval of interest (in this case, t=0 to t=10) into smaller intervals and approximating the solution at each interval using the derivative at that point. The approximation is then used to calculate the solution at the next interval, and this process is repeated until the desired endpoint is reached.

To approximate k using the shooting method and Euler's method, we can follow these steps:

1. Choose an initial guess for k, let's call it k0.
2. Using Euler's method, solve the differential equation dy/dt = k0y from t=0 to t=10, with the initial condition y|t=0 = 1. This will give us an approximation of y at t=10, let's call it y10.
3. Compare y10 to the desired boundary condition of y|t=10 = 4. If they do not match, adjust the value of k0 and repeat step 2 until a satisfactory approximation is found.
4. Once a satisfactory approximation is found, the value of k used in step 2 is the approximate value of k for the given differential equation.

In summary, the shooting method and Euler's method can be used together to approximate the value of a constant in a differential equation. By adjusting the initial guess for the constant and using Euler's method to solve the equation, we can iteratively find a satisfactory approximation.