九六期末第三题的B B) Let f'(x) = kf(x) for all x in some interval. Prove that f(x) = Ce^kx , where C is an arbitrary constant. 两个证明方法 (M1) set f(x)=ce^kx => f'(x)=kce^kx=k*f(x) (M2) 这题其实是Seperable ODE set y=f'(x) we can rewrite the equation as follow dy ---- = k y dx seperate x y dy ---- = k dx y intergrate both sides ln y = kx + C => y= e^(kx+C) = e^kx * e^C = C' e^kx (((C'=e^c))) 然後cross section perpendicular to the x-axis就是截面积跟x轴垂直 绕着X轴旋转的意思 以上 --

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