9. [25 pts.] Consider the following ODE for y(x): y' + sin(x + y) = 0....

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9. [25 pts.] Consider the following ODE for y(x): y' + sin²(x + y) = 0. (1) (a) What is the order of this ODE? Is it linear or nonlinear? Separable or non-separable? (b) What can you say from general theorems about the existence and unique- ness of solutions of the initial value problem for (1) with the initial condition y (xo) = yo? (c) Show that the substitution u(x) = x + y(x) gives the following ODE: u' cos² u. = 2 (d) Solve (2) for u(x) and write down the general solution of (1) for y(x). For what x-values do your solutions exist? (e) What are the solutions of (1) with the initial conditions: (i) y(0) = 0; (ii) y(0) = π/2? 9. [25 pts.] Consider the following ODE for y(x): y' + sin²(x + y) = 0. (1) (a) What is the order of this ODE? Is it linear or nonlinear? Separable or non-separable? (b) What can you say from general theorems about the existence and unique- ness of solutions of the initial value problem for (1) with the initial condition y (xo) = yo? (c) Show that the substitution u(x) = x + y(x) gives the following ODE: u' cos² u. = 2 (d) Solve (2) for u(x) and write down the general solution of (1) for y(x). For what x-values do your solutions exist? (e) What are the solutions of (1) with the initial conditions: (i) y(0) = 0; (ii) y(0) = π/2?

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