$(1)$

$d\frac{y}{d}x=x^2-y^3$

$y(0)=0$

le noticed before that there is no way for us to solve this system by any analytic method, but since $x^2{y}^3$ is a continuous function, we know that a solution exists on some interval around the initial value $x0.$ We saw using slope field and drawode roughly what the solution curve looks like. Now let's se our new numerical method to estimate the solution.

Exercise $3\mathrm{.}3$

a$.$ Run Euler's Method in MATLAB for the inital value problem $(1)$ on the interval $0x4.$ Use different colors to plot the estimates for the following values of $h$:$1,0\mathrm{.}5,0\mathrm{.}25,$ and $0\mathrm{.}1.$

Remember to use hold on$!$ Some of the colors you can use with MATLAB's plot command are $\text{'}r\text{'},\text{'}g\text{'},\text{'}b\text{'},\text{'}c\text{'},\text{'}$m$\text{'},\text{'}$y$\text{'},$ and $\text{'}k\text{'}.$

Notice that the $y-$axis is very large. Rescale the $y-$axis to range from $-10$ to $10.($Assignment $1$ showed us how to do this.$)$ Given the results of the plot, what can you say about the accuracy of the $h=0\mathrm{.}1$ case? You do not need to include the actual data in the table $x,y$ in your writeup, only the plot and your written answer.

b$.$ Enter hold off so that MATLAB will overwrite your previous plot, and this time, run Euler's Method with the $h$ values $1,0\mathrm{.}1,0\mathrm{.}01,$ and $0\mathrm{.}001.$ Plot the results on one graph using colors of your choice, and rescale so that $y$ ranges from $0$ to $2\mathrm{.}5.$ Paste this plot into your writeup.

Did the additional shrinking of $h$ change the results significantly? What does this suggest about the accuracy of your solution for each step size? Comment on the consistency of your plot with the direction field shown at the top of the page, and also describe any inconsistencies between different step sizes.

Answer in MATLAB code please!