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This leads to the equation &Newton s second lawZwhere m is the mass of the object, h is its height above the ground, d2h / dt2 is its acceleration, g is the gravitational constant, and ( mg ) is the force due to gravity.!' Meaning of the DEqmWhat do we mean by a differential equation? An equation containing derivatives of the unknown function. 2,A,BExamples  Examples: ------------- (1) where a and k are parameters or coefficients -------------- (2)  Z|ZZ?ZZ}ZZ`Z i33,Pn |,_Examples   --------------- (3) --------------- (4) --------------- (5) ??k+$b,b$m b  b $$D+B V #Examples  P --------------- (6) ---------------- (7) ---------------- (8)2QqbY`* o$Order and degree "\The order of the DEq: is the order of the highest-order derivative which appears in the equation. % Eq (1) is a second order, whereas Eq (2) is a first order. The degree of the DEq: is the degree of the highest derivative in the DEq. % Eq (4) is of second degree, whereas Eq (5) is of first degree.vcZ?ZLZAZb<N?R *1"% (Classification of Differential Equations))' Ordinary differential equations (ODEqs): involving ordinary derivatives w.r.t. a single independent variable (for example, all above Eqs. except Eq (2)). Partial differential equations (PDEqs): involving partial derivatives w.r.t more than one independent variable (for example, only Eq (2)).2PPP(tI8 )!7& Types of ODEqs (1)Two types of ODEqs: % Linear ODEq: Any equation that can be written in the form -------(A) where and F(x) depend only on the independent variable x (not on y) PPPEP&-F   0 8* Types of ODEqs (2)$ % Nonlinear ODEq: if it is not linear. For example, Eq (6) is a linear second order DEq, whereas Eq (7) and Eq (8) are nonlinear. Question: As written, indicate whether the following DEq is linear or nonlinear. <rHx5  5+ Solutions of ODEqs (1)$ \% Any nth order ODEq can be expressed in the general form -------------------- (B) where F is a function of the independent variable x, the dependent variable y, and the derivatives of y up to order n./A9|b+ *:| x1 Solutions of ODEqs (2)$ We shall assume that it is always possible to solve a given ODEq for the highest derivative, obtaining, --------------------(C) That is, we assume that x lies in an interval I that can be any o the usual intervals (a, b), [a, b], [a, b), and so on.^iZuyi`4hy2Solutions of ODEqs (3)Definition: By a solution of the ODEq (B) or (C) on the interval a < x < b we mean a function such that for every x in (a, b) exist and satisfy or [ 6'``6`b%`Z 7 Examples (1)    PExample 1: Show that = exp(3x) is a solution to on the interval Solution: The given function and its derivative are defined on When we substitute them into the given DEq, we find 3exp(3x) = (3) (exp(3x)), which is true for all x in Hence = exp(3x) is a solution to the given DEq. Q ```b6`b'b`ebb$bbbb  T  a  < Examples (2)   Example 2: Show that =x2  x-1 is a solution to Solution: The given function and its first and second derivatives are defined on and . When we substitute them into the given DEq, we find *ZZ ```h`h```H    A Examples (2a)  Since this is valid for any x `" o, the function = x2  x-1 is a solution to the given DEq. Question 1: Given an equation of the form (C), how do we know whether it even has a solution? This is the question of the existence of a solution.zB`h`h`! <J F Examples (3)   Example 3: The DEq has no real-valued solution since the left side of the DEq is positive for all real-valued functions which are differentiable at the real variable. Question 2: Assuming a given equation has one solution, does it have other solutions? If so, what type of additional conditions must be satisfied in order to single out a particular solution? This is the question of uniqueness.  bbbb  t  ?  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