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生存方程部分精确求解
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[frac{{{rm{dlnB}}}}{{{rm{dt}}}}{rm{ = X}}]
则可以得到如下简化的方程：
[frac{{{{rm{d}}^{rm{2}}}{rm{X}}}}{{{rm{d}}{{rm{t}}^{rm{2}}}}} + {rm{M}}{{rm{e}}^{rm{X}}}{rm{ - Nt = 0}}]
如果不考虑自然衰减的因素，则e1和e2皆为0，方程可以简化为（以下推导过程中C, D, F皆为任意常数）：
frac{{{{rm{d}}^{rm{2}}}{rm{X}}}}{{{rm{d}}{{rm{t}}^{rm{2}}}}} + {rm{M}}{{rm{e}}^{rm{X}}}{rm{ = 0}} \
P = frac{{dX}}{{dt}} \
frac{{{{rm{d}}^{rm{2}}}{rm{X}}}}{{{rm{d}}{{rm{t}}^{rm{2}}}}} = frac{{dP}}{{dt}} = frac{{dP}}{{dX}}frac{{dX}}{{dt}} = Pfrac{{dP}}{{dX}} \
Pfrac{{dP}}{{dX}} = - M{e^X} \
PdP = - M{e^X}dX \
d(frac{1}{2}{P^2}) = d( - M{e^X}) \
P = pm sqrt {2(Const. - M{e^X})} \
end{array}]
frac{{dX}}{{dt}} = pm sqrt {2(Const. - M{e^X})} \
frac{{dX}}{{ pm sqrt {2(Const. - M{e^X})} }} = dt \
pm D{rm{[}}{{rm{e}}^{ - frac{{rm{X}}}{{rm{2}}}}}{({{rm{e}}^{{rm{ - X}}}} - {rm{C}})^{ - frac{1}{2}}}{rm{]}}dX = dt \
mp {rm{2}}D{rm{[}}{({{rm{e}}^{{rm{ - X}}}} - {rm{C}})^{ - frac{1}{2}}}{rm{]}}d{{rm{e}}^{{rm{ - }}frac{{rm{X}}}{{rm{2}}}}} = {rm{dt}} \
{{rm{e}}^{{rm{ - }}frac{{rm{X}}}{{rm{2}}}}} = {rm{Y}} \
{rm{Y}} = frac{1}{{sqrt {{{rm{e}}^{{rm{lnB}}}}} }} = frac{1}{{sqrt B }} \
mp {rm{2}}D{({{rm{Y}}^2} - {rm{C}})^{ - frac{1}{2}}}d{rm{Y}} = {rm{dt}} \
end{array}]
mp {rm{2}}D(ln |Y + sqrt {{{rm{Y}}^2} - {rm{C}}} | + Const.) = {rm{t}} + {rm{F}} \
ln |Y + sqrt {{{rm{Y}}^2} - {rm{C}}} | = pm frac{{t + {rm{F}}}}{{2D}} - Const. \
|Y + sqrt {{{rm{Y}}^2} - {rm{C}}} | = exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.) \
Y + sqrt {{{rm{Y}}^2} - {rm{C}}} = exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.) \
{{rm{Y}}^2} - {rm{C}} = {{rm{[}}exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.){rm{ - Y]}}^{rm{2}}} \
end{array}]
[begin{array}{l}
- {rm{C}} = exp {rm{[2}}( pm frac{{t + {rm{F}}}}{{2D}} - Const.){rm{] - 2Y}}exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.) \
{rm{Y}} = frac{{exp {rm{[2}}( pm frac{{t + {rm{F}}}}{{2D}} - Const.){rm{]}} + {rm{C}}}}{{{rm{2}}exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.)}} \
end{array}]
[begin{array}{l}
{e^X} = {left{ {frac{{exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.)}}{{{rm{2}}exp {rm{[2}}( pm frac{{t + {rm{F}}}}{{2D}} - Const.){rm{]}} + {rm{C}}}}} right}^2} \
B(t) = {left{ {frac{{exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.)}}{{{rm{2}}exp {rm{[2}}( pm frac{{t + {rm{F}}}}{{2D}} - Const.){rm{]}} + {rm{C}}}}} right}^2} \
end{array}]
[B(t) = {left{ {frac{{rm{1}}}{{{rm{2}}exp ( pm frac{{t + {rm{F}}}}{{2D}} - Const.) + {rm{C}}exp {rm{[ - }}( pm frac{{t + {rm{F}}}}{{2D}} - Const.){rm{]}}}}} right}^2}]
则在t --> ∞时，
https://blog.sciencenet.cn/blog-361477-571456.html
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