dw/dt=k(w-c)

代入上述公式，则有

c=∫_0~t wf(w) k(w-c)dt

1=∫_0~t f(w) k(w-c)dt

S=∫_0~T f(w) k(w-c)dt

而

w-c=(w0-c)exp(kt)

将其代入可得

c=∫_0~t wf(w) k(w0-c)exp(kt) dt =∫_0~t wf(w) (w0-c) d(exp(kt))

1=∫_0~t f(w) k(w0-c)exp(kt) dt =∫_0~t f(w) (w0-c) d(exp(kt))

S=∫_0~T f(w) k(w0-c)exp(kt) dt =∫_0~T f(w) (w0-c) d(exp(kt))

设

p=exp(kt)

则有

c=∫_1~p wf(w) (w0-c) dp

1=∫_1~p f(w) (w0-c) dp

S=∫_1~exp(kT) f(w) (w0-c) dp