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20_4 - Solving Differential Equations 20.4 Introduction In...

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Solving Differential Equations 20.4 Introduction In this Block we employ the Laplace transform to solve constant coefficient ordinary differential equations. In particular we shall consider initial value problems. We shall find that the initial conditions are automatically included as part of the solution process. The idea is simple; the Laplace transform of each term in the differential equation is taken. If the unknown function is y ( t ) then, on taking the transform, an algebraic equation involving Y ( s ) = L{ y ( t ) } is obtained. This equation is solved for Y ( s ) which is then inverted to produce the required solution y ( t ) = L 1 { Y ( s ) } . Prerequisites Before starting this Block you should ... understand how to find Laplace transforms of simple functions and their derivatives be able to find inverse Laplace transforms using a variety of techniques understand what an initial-value problem is Learning Outcomes After completing this Block you should be able to ... solve initial-value problems using the Laplace transform method Learning Style To achieve what is expected of you ... allocate sufficient study time briefly revise the prerequisite material attempt every guided exercise and most of the other exercises
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1. Solving Differential Equations using the Laplace Transform We begin with a straightforward initial value problem involving a first order constant coefficient differential equation. Let us find the solution of d y d t + 2 y = 12e 3 t y (0) = 3 using the Laplace transform approach. Although it is not stated explicitly we shall assume that y ( t ) is a causal function (we have no interest in the value of y ( t ) if t < 0). Similarly, the function on the right-hand side of the differential equation (12e 3 t ), the ‘forcing function’, will be assumed to be causal. (Strictly, we should write 12e 3 t u ( t ) but the step function u ( t ) will often be omitted). Let us write L{ y ( t ) } = Y ( s ). Then, taking the Laplace transform of every term in the differential equation gives:
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