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**Problem I.**

Consider the equation

[tex] y''+a_1y'+a_2y=0[/tex]

where the constants [tex] a_1,a_2[/tex] are real. Suppose [tex] \alpha+i\beta[/tex] is a complex root of the

characteristic polynomial, where [tex] \alpha,\beta[/tex] are real, [tex] \beta\neq 0[/tex] .

(i) Show that [tex] \alpha-i\beta[/tex] is also a root.

(ii) Show that any solution [tex] \phi[/tex] may be written in the form

[tex] \phi(x)=e^{\alpha x}(d_1\cos(\beta x)+d_2\sin(\beta x))[/tex] ,

where [tex] d_1,d_2[/tex] are constants.

(iii) Show the [tex] \alpha=-a_1/2, \beta^2=a_2-(a_1^2/4)[/tex] .

(iv) Show that every solution tends to zero as [tex] x\rightarrow +\infty[/tex] if [tex] a_1>0[/tex] .

(v) Show that the magnitude of every non-trivial solution assumes aribtrarily large values as [tex] x\rightarrow +\infty[/tex] if [tex] a_1<0[/tex] .

**Problem II.**

Show that every solution of the constant coefficient equation

[tex] y''+a_1y'+a_2y=0[/tex]

tends to zero as [tex] x\rightarrow \infty[/tex] if, and only if, the real parts of the roots of the characteristic polynomial are negative.

I cannot seem to solve parts (iii) through (v) in problem I and the whole of problem II correctly. I am really not all that sure where I should begin on these parts. Any assistance offered would be much appreciated. Thank you for your time.