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### Separation of variables

This technique can be used to solve more complicated equations, such as those where the rate of change is a product of functions of the dependent and of the independent variables, i.e. To solve such an equation, we first separate the variables'' as shown. We now integrate both sides to obtain the solution We then proceed to find the particular solution, as before.

As an example, we will use the equation, arising in physics, modelling radioactive decay. Here, is a constant known in physics as the decay constant''. represents the number of undecayed nuclei, and is the time in seconds. The equation is (The negative sign shows that is decreasing.) To solve the equation, we separate the variables to obtain We now integrate both sides to find the general solution. Here, represents the initial number of undecayed nuclei - the number at .   Next: Copyright Up: Differential equations Previous: Direct integration
Alexander Frolkin 2001-03-13