One way to define hyperbolic functions is by means of differential equations. Consider t...

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: and The hyperbolic sine, sinh t, is defined as the solution of this equation subject to the initial values: and

(a) Solve these initial value problems to derive explicit formulas for cosh t, and sinh t. Also show that 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 ar ² + br + c = 0 has two distinct real roots. If the two roots are expressed in the form and , show that a general solution of the equation

(d) Use the result of part (c) to solve the initial value problem: -17/2.