Computing approximate Solutions/ Euler's Method

Given the equation dy/dt= t-(y^2) compute 2 approx. solutions corresponding to delta t = 1 and delta t = 0.5 on the interval [0,1]. I've included a photo of my work and was wondering if someone could check it for me to see if I did it correctly. I think the solution for delta = 1 and the graph is correct but for delta = 0.5 I'm not sure. Thank you

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Re: Computing approximate Solutions/ Euler's Method

Re: Computing approximate Solutions/ Euler's Method

Hi

Your results are correct.

But, personnally, here is how I would write the answer (I do not be in the habit of making tables).

If you set $t_0=0$, $t_1=0.5$, $t_2=1$ and $y_0=y(0)$, $y_1=y(0.5)$, $y_2=y(1)$, (explicit) Euler's method consists in calculating $y_1$ and $y_2$ from the following equation :

$$

y_{i+1}=y_i+(t_{i+1}-t_i).(t_i-y_i^2)

$$

Since $y_0=y(0)=0$, we thus obtain

$$

y_1=y(0.5)=0.5

$$

and

$$

y_2=y(1)=0.625

$$

Re: Computing approximate Solutions/ Euler's Method

Quote:

Originally Posted by

**ryotiger** (I do not be in the habit of making tables).

Why not? It's the most clear and concise method of organising all the important pieces of information needed when doing these numerical methods...

Re: Computing approximate Solutions/ Euler's Method

Yes, you are right, I did not say the opposite.

It's just a habit that I took to write in words the calculations to help understanding when I write scientific papers or when I teach. But if only the answer is expected, a table can be sufficient. It's only a personal opinion.