# separation of variables

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, 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:

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

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

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

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.

$$\int \frac{dP}{P}=\int 1.15\mathit{d}t$$ |

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

$$P=C{e}^{1.15t}$$ |

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 |