Question # 1

Prove that if f and g are two functions such that the
derivative of f is g, and the derivative of g is f,
then (f(x)+g(x)) is a constant times the exponential function,
e**x.

Define h(x) = ( f(x) + g(x) )/e**x

Take the derivative of h(x) by the quotient rule.
h'(x)=((f'(x)+g'(x))*e**x-(f(x)+g(x))*e**x))/(e**x)**2
Since f'(x) = g(x) and g'(x) = f(x),
f'(x) + g'(x) = f(x) + g(x).

h'(x) = ((f(x)+g(x))*e**x-(f(x)+g(x))*e**x))/(e**x)**2
h'(x) = ( f(x) + g(x) - f(x) - g(x) ) * e**x / (e**x)**2
h'(x) = 0

Thus h(x) is a constant.
(f(x)+g(x))/e**x is a constant.
(f(x)+g(x)) is a constant times e**x.

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