Transcription of Integration and Differential Equations
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Integration and Differential Equations
2 Integration and Differential EquationsOften, when attempting to solve a Differential equation, weare naturally led to computing one ormore integrals after all, Integration is the inverse of differentiation. Indeed, we have alreadysolved one simple second-order Differential equation by repeated Integration (the one arising inthe simplest falling object model, starting on page 10). Letus now briefly consider the generalcase where Integration is immediately applicable, and alsoconsider some practical aspects ofusing both the indefinite integral and the definite Directly-Integrable EquationsWe will say that a given first-order Differential equation isdirectly integrableif (and only if) itcan be (re)written asdydx=f(x)( )wheref(x)is some known function of justx(noy s ). More generally, anyNth-order differ-ential equation will be said to bedirectly integrableif and only if it can be (re)written asdNydxN=f(x)( )where, again,f(x)is some known function of justx(noy s or derivatives ofy).
26 Integration and Differential Equations Cutting out the middle leaves dy dx = 6x3 + c 1. Integrating this, we have y(x) = Z dy dx dx = Z 6x3 +c 1 dx = 6 4 x4 + c 1x + c 2. So the general solution to equation (2.8) is y(x) = 3 2 x4 + c 1x +c 2. In practice, rather than use the same letter with different subscripts for different arbitrary
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ELEMENTARY DIFFERENTIAL EQUATIONS, Equations, Numerical Methods, Methods, Numerical Differential Equations, Numerical differential, DIFFERENTIAL EQUATIONS, Numerical Methods for Partial Differential Equations, Numerical Methods for Differential Equations, Numerical solution, Numerical, 4 Numerical, DIFFERENTIAL