Problem 2: Consider the following Initial Value Problem (IVP) where (?) is the dependent function:
? ′ = ? − ? 2 + 1.14 cos(? ?/2 ) with ?(0) = ?0 and ? ≥ 0
Part 2A:
Approximate the solution to the IVP using the Improved Euler’s method with the following conditions: Initial condition ?0 = 1; time steps ℎ = 1 8 , 1 16 , 1 32 , 1 64 ; and time interval ? ∈ [0,20]
Plot the Improved Euler’s method approximation for all 4 time steps
Discuss the results of these approximations
? ′ = ? − ? 2 + 1.14 cos(? ?/2 ) with ?(0) = ?0 and ? ≥ 0
Part 2A:
Approximate the solution to the IVP using the Improved Euler’s method with the following conditions: Initial condition ?0 = 1; time steps ℎ = 1 8 , 1 16 , 1 32 , 1 64 ; and time interval ? ∈ [0,20]
Plot the Improved Euler’s method approximation for all 4 time steps
Discuss the results of these approximations