For what values of the constant a can the Fundamental Existe

For what values of the constant a can the Fundamental Existence and Uniqueness Theorem 1 be applied to the initial value problems below? Explain your answer. What do you conclude from this theorem for such values of a? What do you conclude from this theorem for other values of a?

Solution

132.

according to the fundamental existence and uniqueness theorem a differnetial equation shall have a unique solution and for that

x\'=tan(ax)

F(x,t) tan(ax) , ---------(1)

dF/dx = a*sec^(ax) , w----------(2)

both (1) and (2) need to be defined

tan and sec function are not defined for +-pi/2 , +- 3pi2 , .......................

and to be more specific for tan and sec to be defined their domain needs to be beteen

-pi/2 < x < pi/2       ----------(3)

dx/dt = tan(ax)

sec(ax)dx = dt

integrating both sides

=> ln[tan(ax) + sec(ax)]/a = t + C

and x(0) =pi/2

=> C = ln[tan(api/2) + sec(api/2)]

=> the solution of the differential equation becomes

ln[tan(ax) + sec(ax)]/a = t + C = ln[tan(api/2) + sec(api/2)]

its evident that if ax = pi/2 , that is if a = +-pi/2 and x=+-1 then the solution does not exist and if a = +-1 for x = +-pi/2 then the solution does not exist.

hence for all values of \'a\' except +- pi/2 for x=+-1 and +-1 for x=+-pi/2 the fudamental existance and uniquness theorem exists.