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Unformatted text preview: Differential Equations Terminology Definition Let y = f ( x ) . An ordinary differential equation is a relation between y and one or more of its derivatives. I Example The following are all ordinary differential equations: (1) dy dx + 5 xy = 0 (2) d 2 y dx 2 = x + y 2 (3) y 000 + xy + x 2 y = 0 (4) cos( x ) d 2 y dx 2 + sin( x ) dy dx + y = sec( x ) Definition The order of a differential equation is the order of the highest derivative appearing in the equation. I Example Equation (1) is first-order, while equations (2) and (4) are both second order and equation (3) is third-order. Definition A differential equation is linear if it can be written in the form y ( n ) + f n- 1 ( x ) y ( n- 1) + + f 1 ( x ) y + f ( x ) y = g ( x ) , otherwise it is nonlinear . Furthermore, if g ( x ) = 0 then we say the linear differential equation is homogeneous , and otherwise non-homogeneous . I Example Equations (1), (3), and (4) are linear. Notice that equation (2) is nonlinear because of the term y 2 . I...
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