 # 1. (30 points) Consider the IVP ODE: y' = -y+ cos t + 4, to IC: ...

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1. (30 points) Consider the IVP ODE: y' = -y+ cos t + 4, to IC: y(0) = 0 (a) Prove that the ODE is linear by showing that the linearity property holds. (b) Study the existence and uniqueness of the solution to the IVP. (c) Use the method of integrating factor and find the solution to the IVP in explicit form. (d) Describe the behavior of the solution as t 80. (e) Use MATLAB and plot the graph of the solution for t € [0, 100]. 2. (30 points) Consider the IVP ODE: of IC: y(0)=2 = (a) Determine whether the ODE is linear or nonlinear. You do not need to prove it, but you need to motivate your answer. (b) Study the existence and uniqueness of the solution to the IVP. (c) Show that the ODE is separable. (d) Find the solution to the IVP in explicit form. (e) Use MATLAB and plot the graph of the solution for I € [0, 10]. 3. (40 points) Consider the IVP ODE: y = -(42/x) 2 y Inz IC: y(2)=1 = (a) Determine whether the ODE is linear or nonlinear. You do not need to prove it, but you need to motivate your answer. (b) Study the existence and uniqueness of the solution to the IVP. (c) Show that the ODE is exact. (d) Find the solution to the IVP in both implicit and explicit forms. (e) (Extra 10 bonus points) Use MATLAB and plot the graph of the solution for x E (2,5], using both the implicit and explicit forms of the solution. Note that when you illustrate the solution based on its implicit form, you need to use contour command, and when you illustrate the solution based on its explicit form, you need to use plot command. I have put a sample code on the course webpage. You can modify and use it for this problems.

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