Applications of Hyperbolic Functions

STEP 2001, Math III, #2: Show that

Show that the area of the region defined by the inequalities and is .

STEP 2005, Math III, #6: In this question, you may use without proof the results

Show that the equation is satisfied by

then one of the roots of the equation is , where .

Show that the other roots of the equation are the roots of the quadratic equation

, and , where

Solve completely the equation .

STEP 2006, Math III, #7: (i) Solve the equation giving in terms of .

Find the solution of the differential equation

and at .

(ii) Find the solution, not identically zero, of the differential equation

at , expressing your solution in the form . Show that the asymptotes to the solution curve are .

STEP 2007, Math III, #5: Let , where , and let and be functions of determined by and . Show that

and for .

Find, with proof, a similar formula for in terms of and .

STEP 2008, Math III, #4: (i) Show, with the aid of a sketch, that for and deduce that

(*)

for

.

(ii) By integrating (*) , show that

for

.

(iii) Show that

for

.