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**Question**

**a) Find two linearly independent solutions of [itex] t^2x''+tx' - x = 0[/itex]**

**b) Calculate Green's Function for the equation [itex] t^2x''+tx' - x = 0[/itex], and use it to find a particular solution to the following inhomogeneous differential equation.**

[tex]t^2x''+tx'-x = t^4[/tex]

**c) Explain why the global existence and uniqueness theorem guarantees that, if [itex]f:(0,\infty)\rightarrow \mathbb{R}[/itex] is continuous, the the initial value problem**

[tex] t^2x''+tx' - x = f(t), \quad \quad x'(1) = x(1) = 0[/tex]

**has a unique solution on [itex](0,\infty)[/itex]. Find an example of a continuous function [itex]f:(0,\infty) \rightarrow \mathbb{R}[/itex] such that the solution of the above IVP satisfies [itex]|x(t)|\rightarrow \infty[/itex] as [itex]t\rightarrow 0+[/itex], so that the solution is not continuous on [itex][0,\infty)[/itex].**