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**Unformatted text preview: **August 30, 2011 2-1 2. First Order Linear Equations and Bernoullis Differential Equation First Order Linear Equations A differential equation of the form y + 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 : < t < } . 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 ( , ), then, for any real number y , there is a unique solution y ( t ) to the initial value problem y + p ( t ) y = g ( t ) , y ( t ) = y (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 such that ( y ) = g Then, we would have y + y = g y + ( g- py ) = g y = py = p d log dt = p August 30, 2011 2-2 Since p = p ( t ) is a continuous function of t , we can integrate both sides to find log , and then take the exponential to find ....

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