Question Regarding Initial Conditions.

[abcd200x]

I was doing a question which involved using euler's method to numerically integrate the equation dy/dt=2ty^2 in the interval [0,1], with h=0.2 but no initial conditions were given. How do I find them. I know it's something simple but I can't get it.

Thanks

 

[arildno]

You cannot "find" an initial condition if you only have a differential equation to begin with.
The initial condition is an arbitrary parameter which secures the uniqueness of a solution to a differential equation problem. Without it, you have, in general, an infinity of solutions (the general solution).

However, knowing the general solution can give you a good hint as to which initial conditions may cause you trouble in a numerical procedure.

In your case, the general solution consists of the following functions:
Y1(t)=0
Y2(t)=-1/(C+t^(2)), C not eq. 0.

Y1(t) is the unique solution to the problem with init. cond. y(0)=0,
while Y2(t) yields the unique solution for any other choice of y(0).

Note however, that if y(0) is positive, C will be negative, and there will exist two singularities in your solution, at t1=-sqrt(-C), t2=sqrt(-C).

For a numerical integration with an arbitrary parameter y(0) included, this will be destructive.
The most general and safe way to proceed is to assume y(0)=-a^(2), where a is a real number.

 

[M98Ranger]

for reaching out with your question! In order to use Euler's method to numerically integrate a differential equation, we do need to have initial conditions. These initial conditions represent the starting point for our integration and help us determine the value of the solution at each step. In this case, since no initial conditions were given, you will need to make assumptions or estimates for the initial values of y and t in order to proceed with the integration.

One approach could be to assume a value for y at t=0, and then use that to calculate the value of y at t=0.2, 0.4, 0.6, and so on until you reach t=1. You can then compare your calculated values with the actual solution at those time points to see how close your initial assumption was. If it is not accurate enough, you can adjust your initial assumption and repeat the process until you are satisfied with the results.

Another approach could be to use a graphing tool or software to plot the differential equation and visually estimate the initial values of y and t based on where the solution curve intersects the t-axis at t=0. This can give you a rough estimate to start with and then you can refine it as needed.

I hope this helps and good luck with your problem!