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The dependent variable y and independent variable t in Eq.(1) can be separated to each side of the equation:dy/y = -kdt ---------- (2)
Then it can be solved by integration:
Figure 01 Integration [view large image]
|where is a symbol indicating integration. For example, the first term in Eq.(3) represents the sum of the small strips under the curve 1/y with dy 0 (Figure 01). In term of the natural log "ln" (the 2nd term in Eq.(3), see derivation in Derivation of some Differential Formulas), these integrals can be expressed in the analytical form:|
where e = (1 + 1/n)n 2.71828... as n is the exponential base used to simplify the form of many formulas (it plays a special role in calculus similar to the "" in geometry), "ln" is the logarithms in this base, and A is the integration constant (which vanishes and the original differential equation is recovered if we reverse the process by taking the differentiation again) to be determined by the initial condition.
For example, if at t = 0, y = yo, then A = yo. Alternatively, the integration constant can be replaced by the definite integral such that f(y) dy = F(y) = F(b) - F(a), where F(y) is the result of the integration.
In either way, the final solution for Eq.(2) is :
y = yo e-kt --------------------------------- (4)
The green curve in Figure 02 shows the radioactive decay of Iodine 131 with a half life of 8.1 days. Half life is the interval when half of the original is gone. This graph corresponds to a value of 1/k = 8.1/loge(2) = 11.57 days, and yo = 1 gm. in Eq.(4).
Figure 02 Radiactive Decay [view large image]