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separation of variables
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Separation of variables is a valuable tool for solving differential equations of the form $$\frac{dy}{dx}=f(x)g(y)$$ The above equation can be rearranged algebraically through Leibniz notation to separate the variables and be conveniently integrable. $$\frac{dy}{g(y)}=f(x)dx$$ It follows then that $$\int\frac{dy}{g(y)} = F(x) + C$$ where $F(x)$ is the antiderivative of $f$ and $C$ is a constant of integration. This gives a general form of the solution. An explicit form may be derived by an initial value.
Example: A population that is initially at $200$ organisms increases at a rate of $15\%$ each year. We then have a differential equation $$\frac{dP}{dt} = 0.15P$$ The solution of this equation is relatively straightforward, we simple separate the variables algebraically and integrate. $$\int \frac{dP}{P} = \int 0.15\;dt$$ This is just $\ln P = 0.15t + C$ or $$P=Ce^{0.15t}$$ When we substitute $P(0)=200$ , we see that $C=200$ . This is where we get the general relation of exponential growth $$P(t) = P_0e^{kt}$$
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"separation of variables" is owned by slider142.
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Cross-references: exponential growth, relation, integrate, simple, solution, constant of integration, antiderivative, variables, Leibniz notation, equation, differential equations
There are 18 references to this entry.
This is version 2 of separation of variables, born on 2002-02-26, modified 2002-02-26.
Object id is 2712, canonical name is SeparationOfVariables.
Accessed 12717 times total.
Classification:
| AMS MSC: | 34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions) | | | 34A09 (Ordinary differential equations :: General theory :: Implicit equations, differential-algebraic equations) | | | 34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general) |
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Pending Errata and Addenda
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