1. Consider the following differential equation for y(x) and F(x) dy dy dx dx +2a- +9ay...

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1. Consider the following differential equation for y(x) and F(x) d²y dy dx dx² +2a- +9a²y = F(x) (1) where a is some arbitrary non-zero real constant and y(0) = 3 and y'(0) = a. (a) For which values of a does y(x) exhibit oscillations? (b) Solve equation (1) where F(x) = 0. =e-ax (c) Solve equation (1) where F(x) Check that your final solution indeed satisfies y(0) = 3 and y'(0) : = a. If not, you have made a mistake! (d) Solve equation (1) where F(x) = = e-a* sin(2√√/2ax + R), for some real constant R. (e) Part (d) yields more than one particular solution. Is your final solution to part (d) unique? Discuss briefly (no more than 6 lines). There is no need to prove your result. 1. Consider the following differential equation for y(x) and F(x) d²y dy dx dx² +2a- +9a²y = F(x) (1) where a is some arbitrary non-zero real constant and y(0) = 3 and y'(0) = a. (a) For which values of a does y(x) exhibit oscillations? (b) Solve equation (1) where F(x) = 0. =e-ax (c) Solve equation (1) where F(x) Check that your final solution indeed satisfies y(0) = 3 and y'(0) : = a. If not, you have made a mistake! (d) Solve equation (1) where F(x) = = e-a* sin(2√√/2ax + R), for some real constant R. (e) Part (d) yields more than one particular solution. Is your final solution to part (d) unique? Discuss briefly (no more than 6 lines). There is no need to prove your result.