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# Diff Eq

### definitions for differential equations terms

Question | Answer |
---|---|

IVP | A differential equation with initial values is called an initial value problem: y(x_0)=c_0,y'(x_0)=c_1,...,y^(n-1)(x_0)=c_(n-1) |

Homogeneous Equation | If f(&x,&y)=&^kf(x,y), then f is said to be homogeneous of degree k, where k is a real number. |

differential equation | An equation that contains derivatives of differentials of one or more variables is said to be a differential equation. |

dependent variable | A dependent variable is a function of an independent variable (allowed to vary on its own). |

parameters | Parameters are quantities allowed to vary beside the variables, but are not either independent or dependent. |

order of diff eq | The order of a differential equation is the order of the highest-ordered derivative in the equation (not the degree). |

general form of nth order diff eq | F(x,y,dy/dx,d2y/dx2,...,dny/dxn)=0 |

solution of diff eq | y_1 is a solution of F(x,y,...,dny/dxn)=0 if: (a) y_1 is n-times differentiable (b) F(x,y_1,...,dny_1/dxn)=0 |

explicit solution | A solution in explicit form is called an explicit solution (the dependent variable is written explicitly as a functioin of the independent variable(s) only). |

implicit solution | g(x,y)=0 is an implicit solution if it defines at least one real explicit solution to the differential equation on a given interval |

particular solution | A particular solution is a solution without any arbitrary constants; these arbitrary constants are plugged in for. |

general solution | A general solution is a solution form which every solution can be derived (by choosing arbitrary constants). |

singular solution | A solution that cannot be obtained by giving values to arbitrary constants in a family of solutions is called a singular solution |

autonomous equation | y'=f(y) where f, the driving function, is a function of the dependent variable only is called an autonomous equation. The independent variable does not appear explicitly. |

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efgray