8/2/2019 Differential Equations Definitions

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Definitions

1. Bernoulli Equation (1st Order ODE): equation in the form of a linear equation but having anadditionaly

nterm

a. Changed to linear form by substitution ofyn variable for solvingi.

b. Bernoulli Form: 2. Exact Equation (1st Order ODE): equation in which the partial differentiation of separate parts

the general form by the separate variables, respectively, affords a single function

a. Can only be solved by integrating!b. Exact Form: () ()

i. and ii. OR

3. Explicit Function: a function written in the usual form of a formula applied to an independentvariable.

a. is explicitb. i.e. independent variable (y) is function of other variable (x))

4. First-Order (Ordinary) Differential Equation: an equation having highest ordered derivativebeing a single derivative

a. dy and dx are differentialsb. Ordinary Differential Equation (ODE):c. Can only be solved by conversion to exact form and integrating!

i. May need a change of variables via Bernoulli or Homogeneous methods5. General Solution: a set of solutions to a differential equation with as many arbitrary constants

(C) as the order of the equation to solve for dependent variable

a. is the general solution to 6. Homogeneous Form (1st Order ODE): a function in which the dependent and independent

variables are directly proportional to one another within the differential equation

a. Homogeneous Form: ()b. Requires change of variables to solve

i. 7. Implicit Function: a function that exists but may be difficult or impossible to write; a relation

(equaling a constant) generates a function to solve the equation

a. and are implicit8. Initial Value Problem: problem in which the solution to a differential equation takes a unique,

specified value at a given point.

8/2/2019 Differential Equations Definitions

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Definitions

a. is the uniquesolution to the initial value problem with and for initial value of1

9. Integrating Factor: key factor for first-order linear ODEs to be in exact forma. Factor leading linear DEs to be exact (by being multiplied to all terms) b.

10.Linear Equation (1st Order ODE): treating ordered variables by derivative as youwould ordered variables by power, linear equations take a form analogous to the

algebraic quadratic formula

a. Converted to exact form via addition of integrating factor to solve!b. Quadratic Formula: c. Linear Form: ()

i. Written as: 11.Order: the highest order of derivative that occurs in the equation12.Particular Solution: a function that satisfies the equation (proper/not arbitrary)

a. is aparticular solution to 13.Separable Equation: equation in which the two variables can be separated into two terms

a. Each term in exact form to be solved!b. i.e. all of the xs on one side of the equation and all the ys on the other

i. => 14.Singular Solution: a particular solution to an equation which is not an instance of the general

solution