EDE 0 Definitions 2014-03-13

Ordinary Differential Equation The unknown function depends on a single independent variable

Partial Differential Equation The unknown function depends on several independent variables and the derivatives are partial derivatives

Order The order of a differential equation is the order of the highest derivative that appears in the equation

Linear Equations The ordinary differential equation 𝐹𝐹�𝑡𝑡,𝑦𝑦,𝑦𝑦′ , … , 𝑦𝑦(𝑛𝑛)� = 0

is said to be linear if F is a linear function of the variables �𝑦𝑦, 𝑦𝑦′ , … ,𝑦𝑦(𝑛𝑛)� or can be written in the form

𝑎𝑎0(𝑡𝑡)𝑦𝑦(𝑛𝑛) + 𝑎𝑎1(𝑡𝑡)𝑦𝑦(𝑛𝑛−1)+. . . +𝑎𝑎𝑛𝑛(𝑡𝑡)𝑦𝑦 = 𝑔𝑔(𝑡𝑡)

Nonlinear Equations Equations that cannot be written in the form above like those with terms yy’, y²y’, sin[y], etc…

Characteristic Equation 𝑎𝑎𝑎𝑎2 + 𝑏𝑏𝑎𝑎 + 𝑐𝑐 = 0

Wronskian Wronskian is the denominator of the c1 and c2 of the general solution to 2nd order linear equations. Wronskian is used to know if the denominator is non-zero. This will determine that c1 and c2 exists.

fundamental set of solutions y1 and y2 form a fundamental set of solutions if the Wronskian is non-zero

and when they are combined, they provide the general solution of the differential equation

general solution the expression that contains all possible solutions Explicit Function Grapher http://www.onlinefunctiongrapher.com/ Implicit Function Grapher http://www.flashandmath.com/mathlets/calc/implicit/implicit.html Direction Field Plotter http://www.dartmouth.edu/~rewn/dfDemo.html Differential Equation Solver http://www.wolframalpha.com/widgets/view.jsp?id=e602dcdecb1843943960b5197efd3f2a

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