# separation of variables

 $\frac{dy}{dx}=f(x)g(y)$

The above equation can be rearranged algebraically through Leibniz notation, treating dy and dx as differentials  , to separate the variables and be conveniently integrable on both sides.

 $\frac{dy}{g(y)}=f(x)dx$

Instead of using differentials, we can also make use of the change of variables theorem for integration and the fact that if two integrable functions are equivalent     , then their primitives differ by a constant $C$. Here, we write $y=y(x)$ and $\frac{dy}{dx}=y^{\prime}(x)$ for clarity. The above equation then becomes:

 $\frac{y^{\prime}(x)}{g(y(x))}=f(x)$

Integrating both sides over $x$ gives us the desired result:

 $\int\frac{y^{\prime}(x)}{g(y(x))}dx=\int f(x)dx+C$

By the change of variables theorem of integration, the left hand side is equivalent to an integral in the variable $y$:

 $\int\frac{dy}{g(y)}=\int f(x)dx+C$

It follows then that

 $\int\frac{dy}{g(y)}=F(x)+C$

where $F(x)$ is an antiderivative of $f$ and $C$ is the constant difference  between the two primitives. 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

 $\frac{dP}{dt}=P+0.15P=1.15P$

The solution of this equation is relatively straightforward, we simply separate the variables algebraically and integrate.

 $\int\frac{dP}{P}=\int 1.15\;dt$

This is just $\ln P=1.15t+C$ or

 $P=Ce^{1.15t}$

When we substitute $P(0)=200$, we see that $C=200$. This is where we get the general relation  of exponential growth

 $P(t)=P_{0}e^{kt}$
 Title separation of variables Canonical name SeparationOfVariables Date of creation 2013-03-22 12:29:24 Last modified on 2013-03-22 12:29:24 Owner slider142 (78) Last modified by slider142 (78) Numerical id 9 Author slider142 (78) Entry type Algorithm  Classification msc 34A30 Classification msc 34A09 Classification msc 34A05 Related topic LinearDifferentialEquationOfFirstOrder Related topic InverseLaplaceTransformOfDerivatives Related topic SingularSolution Related topic ODETypesReductibleToTheVariablesSeparableCase