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lec_2s-an - 2-1 2 First Order Linear Equations and...

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August 30, 2011 2-1 2. First Order Linear Equations and Bernoulli’s Differential Equation First Order Linear Equations A differential equation of the form y 0 + p ( t ) y = g ( t ) (1) is called a first order scalar linear differential equation. Here we assume that the functions p ( t ) , g ( t ) are continuous on a real interval I = { t : α < t < β } . We will discuss the reason for the name linear a bit later. Now, let us describe how to solve such differential equations. There is a theorem which says that under these continuity assumptions, if t 0 ( α, β ), then, for any real number y 0 , there is a unique solution y ( t ) to the initial value problem y 0 + p ( t ) y = g ( t ) , y ( t 0 ) = y 0 (2) which is defined on the whole interval I . Now that we know there is a solution, we can use various methods to try to find it. There is a useful trick (or observation) for this. Assuming y is a non-zero solution to (1), suppose there was a non-zero function μ such that ( μy ) 0 = μg Then, we would have μ 0 y + μy 0 = μg μ 0 y + μ ( g - py ) = μg μ 0 y = μpy μ 0 = μp d logμ dt = p
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August 30, 2011 2-2 Since p = p ( t ) is a continuous function of t , we can integrate both sides to find log μ , and then take the exponential to find μ .
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