One way to define hyperbolic functions is by means of differential equations. Consider the equation y" - y = 0. The hyperbolic cosine, cosh t, is defined as the solution of this equation subject to the initial values: y(0) = 1 and y'(0) = 0. The hyperbolic sine, sinh t. is defined as the solution of this equation subject to the initial values: y(0) = 0 and y'(0) = 1.
a. Solve these initial value problems to derive explicit formulas for cosh t, and sinh t. Also show that d cosh t /dt = sinh t and d sinh t / dt = cosh t.
b. Prove that a general solution of the equation y" - y = 0 is given by y = c1 cosh t + c2 sinh t.
c. Suppose a, b. and c are given constants for which ar2 + br + c = 0 has two distinct real roots. If the