- Solve using Euler’s explicit method the following ODE from x = 0 to x = 2:
��
�� = −1.2� + 7�,-../
with y(0) = 3
a. Use a step size h = 0.5 (solve by hand)
b. Write a script to solve this again with h = 0.05 and h = 0.005. How do you expect the
truncation errors to change?
c. The analytical solution is available in this case and is given by:
�0123 � = 70
9 �,-../ − 43
9 �,8.9/
Calculate the truncation error at x = 0.5, x = 1, x = 1.5, and x = 2 for each of the step sizes.
Plot, compare, and discuss. - Solve using the midpoint method (modified Euler) from x = 0 to x = 2:
��
�� = � −2� +
1
� ��� � ≠ 0
��
�� = 1 ��� � = 0
a. Use a step size h = 0.5 (solve by hand)
b. Write a script to solve this again using a step size h = 0.2
c. The analytical solution is also available in this case and is given by:
�0123 � = ��,/?
Calculate the truncation errors at each x with the step size h = 0.2. Plot and discuss. - Consider the following IVP over x = 0 to 2, and y(0) = 3:
��
�� = 4��,8 − ��
a. Solve by hand using the classic 4th order Runge-Kutta method with an h = 0.5 (it’s tedious,
but I want you to do this just once…it’s a very good exercise for really understanding the
method)
b. Write a script to solve this using 4th order Runge-Kutta method with a step size h = 0.05
c. Use Matlab ode45 to solve this. Compare and discuss.
d. The analytical solution is also available in this case and is given by:
�0123 � = 4 + 5�,/?
Calculate the truncation errors at each x with the step size h = 0.5. Compare with the
truncation errors of the solution using ode45
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