2 consider the 1st order ivp c use eulers method by hand to approximat

Question

2: Consider the 1st Order IVP y^{\prime}=3-2 y-e^{-3 x}, \quad y(0)=1 ) Does a solution to this IVP exist? Is it unique? On what interval does the solution exist?Justify your answers using the existence and uniqueness theorems covered in class. Solve the IVP to find a solution y(x) and write an appropriate interval of definition. c) Use Euler's Method (by hand) to approximate y(0.1), y(0.2), y(0.3), and y(0.4) (use 2 decimalplaces). (d) Use the solution you found in part (b) and the values you found in part (c) to find the absolute error in each approximation performed in (c) (subtract the actual and estimated values and take the absolute value of this quantity). (e) Download the file "eulers_method2.b" from Files -> Written Homework -> Mathematica.Open it in Mathematica and follow the instructions there. (f) Extra Credit: Choose a 1st order IVP of your liking (just make sure the ODE is linear and that a solution exists), modify the Mathematica file to suit this new IVP, run all pieces of code with this new IVP, and submit the modified Mathematica file with your homework.