Differential Equations

First Order Equations

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Learning Curve

Mastering a new area or a skill always takes some time. In this section we try to model the learning process using a differential equation.

First of all, we introduce a measurable learning function \(L\left( t \right).\) This function, for example, may describe the current labour productivity of an employee. Let \({L_{\max }}\) be the maximum available value of \(L\left( t \right).\) In many cases the following thumb rule is valid: the learning speed is proportional to the volume of remaining (unlearned) material. Mathematically, this can be represented by the equation:

\[\frac{{dL}}{{dt}} = k\left( {{L_{\max }} - L} \right),\]

where \(k\) is a coefficient of proportionality. The given differential equation is a separable equation, so it can be easily solved in general form:

\[\frac{{dL}}{{dt}} = k\left( {{L_{\max }} - L} \right),\;\; \Rightarrow \frac{{dL}}{{{L_{\max }} - L}} = kdt,\;\; \Rightarrow \int {\frac{{dL}}{{{L_{\max }} - L}}} = \int {kdt} ,\;\; \Rightarrow - \int {\frac{{d\left( {{L_{\max }} - L} \right)}}{{{L_{\max }} - L}}} = \int {kdt} ,\;\; \Rightarrow - \ln \left( {{L_{\max }} - L} \right) = kt + \ln C,\;\; \Rightarrow \ln \left( {{L_{\max }} - L} \right) = - kt + \ln C,\;\; \Rightarrow \ln \left( {{L_{\max }} - L} \right) = \ln {e^{ - kt}} + \ln C.\]

After eliminating the logarithms, we obtain the general solution in the form:

\[{L_{\max }} - L = C{e^{ - kt}}.\]

The constant \(C\) can be found from the initial condition: \(L\left( {t = 0} \right) = M.\) Hence, \(C = {L_{\max }} - M.\) As a result, the learning curve is described by the formula

\[L\left( t \right) = {L_{\max }} - \left( {{L_{\max }} - M} \right){e^{ - kt}}.\]

The parameter \(M\) is the last expression means the initial level of knowledge or skills. In the simplest case, we can suppose that \(M = 0.\) The other parameter \(k\) controls how fast the curve rises. View of the learning curves at different values of \(M\) and \(k\) is shown in Figure \(1\) and \(2,\) respectively.

Learning curves for different initial levels of knowledge
Figure 1.
Learning curves for different learning rates k
Figure 2.

As it can be seen, the learning level \(L\) in all cases increases in the beginning of the process, and then the learning rate slows down as the level \(L\) approaches the maximum value \({L_{\max }}.\)

See solved problems on Page 2.

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