(a) Show that the solution of the initial value problemy′ = 1 + y2, y(0) = 0is y(x)=tanx....

(a) Show that the solution of the initial value problem

y′ = 1 + y2, y(0) = 0

is y(x)=tanx. (b) Because y(x) = tanx is an odd function with y′(0) = 1, its Taylor series is of the form

y = x + c3x3 + c5x5 + c7x7 +...

Substitute this series in y′ = 1 + y2 and equate like powers of x to derive the following relations:

(d) Would you prefer to use the Maclaurin series formula in to derive the tangent series in part (c)? Think about it!