Homework #9
1. Use Euler’s method with stepsize h= 0.25 to solve the ODE
y0=y+t
for y(1) given y(0) = 1.
2. Consider the ODE
y0=−2ty
with y(0) = 2. The exact solution is y(t)=2e−t2.
(a) Use Euler’s method with stepsize h= 0.5 to approximate y(1) and find the
absolute error E(0.5) of this approximation.
(b) Use Euler’s method with stepsize h= 0.25 to approximate y(1) and find the
absolute error E(0.25) of this approximation.
(c) Compute E(0.5)/E(0.25).
3. Use Midpoint Method with stepsize h= 0.5 to solve the ODE
y0=y+t
for y(1) given y(0) = 1.
4. Use Trapezoid Method with stepsize h= 0.5 to solve the ODE
y0=y+t
for y(1) given y(0) = 1.
5. (Math 274) Consider y0=f(x, y) at x=x2, where x0< x1< x2with regular spacing h.
Integrate both sides from x0to x2. Use Simpson’s Rule to approximate the right hand
side integral. Finally, write down the corresponding implicit approximation scheme for
solving this ODE at x2in terms of data at x0, x1, x2.
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