Consider the following ODE $$ {{dy}\over{dx}}=−\frac{2x}{y}~\text{ where }~ y(0)=1 $$
i) Find the analytical solution.
Answer= (root 1-2x^2)ii) Given that $y(0.7)=0.141421$ to 6 digit precision, use the modified Euler's method to estimate $y(0.8)$ using $h=0.1$ and work to 5 digit precision.
Answer=-0.25928iii) Now use the 4th order Runge-Kutta method to estimate $y(0.8)$. As before, take $y(0.7) = 0.141421$, $h = 0.1$ and work to 5 decimal place accuracy.
Ans=0.07078iv) What does the analytical solution give for $y(0.8)$? Compare the numerical solutions obtained in (ii) and (iii) with this analytical solution and comment on your results. Would you expect better answers to have been obtained if a smaller value of the step size $h$ (and thus more steps) had been used to calculate y(0.8) starting from $y(0.7)$? Explain your reasoning.
Hi, as you can see I've answered the first three parts but don't understand the last part (iv), any help would be appreciated! Sorry for the poor notation, and thanks in advance!