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Mn²⁺ + 4H₂O => MnO^{​​​​​}₄^- + 8H⁺  + 5e^-

x = (-2 +/- sqrt(4-8))/2                    therefore x = - 1 +/- i, where i is the imaginary unit

y² + 2xy dy/dx = 3cos(3x) + 1/y - x/y² dy/dx                   thereforedy/dx = (3cos(3x) + 1/y - y²​​​​​​) / (2xy + x/y²)

Essay Plan [the introduction and conclusion are standard outlines; the paragraphs are based on my own personal answer to the question - a sample answer of sorts].
Intr...