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Unformatted text preview: 1st-Order Linear Homogeneous Case On of the simplest differential equations we can consider is a first-order linear nonhomogeneous differential equation, i.e. one of the form y + f ( x ) y = 0 . A theme we will keep returning to is the idea of an exact derivative. That is, we will try to rewrite part of a differential equation as d dx [ something ] for the simple reason that exact derivatives are easy to integrate. Indeed, we have Z d dx [ something ] dx = something + C. For the moment, suppose y . Then we can rewrite the above equation as y y =- f ( x ) and observe that d dx ln y = y y so now the left hand side is an exact derivative and we have d dx ln y =- f ( x ) . Integrating both sides we get ln y =- Z f ( x ) dx + C and after exponentiating both sides we have y = e- R f ( x ) dx + C = e C e- R f ( x ) dx = C 1 e- R f ( x ) dx . where C 1 = e C . It is not difficult to extend this argument ? to the general case of any continuous function y (not just strictly positive), so that the general solution is given by y = Ce- R f ( x ) dx where C is arbitrary....
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