16b2lec18 - Section 10.1 Solutions of Dierential Equations...

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Section 10.1: Solutions of Differential EquationsAn (ordinary) differential equation is an equation involving afunction and its derivatives.That is, for functionsP(x0, x1, . . . , xn) andQ(x0, . . . , xn) ofn+ 1variables, we say that the functionf(t) (of one variable) satisfiesthe differential equationP(y, y , . . . , y(n)) =Q(f(t), . . . , f(n)(t))ifP(f(t), f(t), . . . , f(n)(t))01ExamplesThe functionf(t) =etsatisfies the differential equationy=y.The constant functiong(t)5 satisfies the differentialequationy= 0.The functionsh(t) = sin(t) andk(t) = cos(t) satisfy thedifferential equationy+y= 0.The function(t) = ln(t) satisfies-(y)2=y.2
3ExampleFind a functionf(t) which satisfiesf(t)f(t) andf(0) = 30.That is, solve the initial value problemy=yandy(0) = 30.4

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