96 Ordinary Differential Equations
c. Estimate the maximum discretization error p er step on the interval[O, l] for the stepsize h = .1.
d. How small must the stepsize be in order to ensure six decimal
place accuracy p er step?- a. Compute a n approximate solution to the initial value problem
y' = x^2 - y; y(O) = 1 on the interval [O, l] using Euler's method
and a const ant stepsize of h = .1.b. Find an upper bound for the total discretization error at x = l.
c. How small must the stepsize be to ensure six decimal place
accuracy per step?
d. How small must the stepsize be to ensure six decimal place
accuracy over the interval [O, l ]?- Use the improved Euler's formula with a stepsize h = .1 to generate a
numerical approximation to the solution of the IVP y' = x^2 -y; y(O) = 1
on the interval [O, l].4. Use the modified Euler's formula with a stepsize h = .1 to generate a
numerical a pproximation to the solution of the IVP y' = x^2 -y; y(O) = 1
on the interval [O, l].5. Use the fourth order Runge-Kutta formula with a stepsize h = .l
to generate a numerical approximation to the solut ion of the IVP y' =x^2 - y; y(O) = 1 on the interval [O, l].
6. a. Find t he exact solution of the initial value problem y' = x^2 - y;
y(O) = l.
b. Compare the va rious approximate solutions generated in exer-
cises 1-5 with each other and the exact solution by producing a
t able of values.- Consider the general recursive formula (16). Suppose that in addition
to satisfying equations (20), we require that the coefficients of j2 fyy in
equations (17) and (19) b e equal. What is the solution of the resulting
syst em of four equa tions in the four unknowns a, b, c, and d? - Generate a numerical solutions to the IVP y' = y/x + 2; y(l) = 1 on
the interval [l, 2] with a stepsize of h = .05 using
a. Euler's method.
b. improved Euler's method.
c. modified Euler's method.
d. the fourth order Runge-Kutta method.