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Unformatted text preview: Wednesday February 2 Lecture 13 : Linear differential equations (Refers to Section 9.5 in your text) Expectations: 1. Recognize a linear first order differential equation. 2. Solve a linear first order differential equation. 13.1 Definition A Linear first-order differential equation is one which can be written in the form We can usually obtain an explicit solution from a linear differential equation rather than implicit solutions that are often obtained with separable DEs. 13.1.1 Examples of linear DEs y ' + 2 y = e x xy ' + 2 y = 4 x 2 y ' + 2 xy = 2 x Note that the third DE is both separable and linear. So there will be two approaches when solving this DE. 13.1.2 We solve a linear equation in two steps : 1. We first obtain the integrating factor where a can be fixed at any suitable value so as to get the simplest function for I ( x ). This is equivalent to taking the anti-derivative of P ( t ) with the most suitable value for the parameter C ....
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