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mws_gen_ode_bck_primer - Chapter 08.01 Primer for Ordinary...

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Chapter 08.01 Primer for Ordinary Differential Equations After reading this chapter, you should be able to : 1. define an ordinary differential equation, 2. differentiate between an ordinary and partial differential equation, and 3. solve linear ordinary differential equations with fixed constants by using classical solution and Laplace transform techniques. Introduction An equation that consists of derivatives is called a differential equation. Differential equations have applications in all areas of science and engineering. Mathematical formulation of most of the physical and engineering problems leads to differential equations. So, it is important for engineers and scientists to know how to set up differential equations and solve them. Differential equations are of two types (A) ordinary differential equations (ODE) (B) partial differential equations (PDE) An ordinary differential equation is that in which all the derivatives are with respect to a single independent variable. Examples of ordinary differential equations include 0 2 2 2 = + + y dx dy dx y d , 4 ) 0 ( , 2 ) 0 ( = = y dx dy , , sin 5 3 2 2 3 3 x y dx dy dx y d dx y d = + + + , 12 ) 0 ( 2 2 = dx y d 2 ) 0 ( = dx dy , 4 ) 0 ( = y Ordinary differential equations are classified in terms of order and degree. Order of an ordinary differential equation is the same as the highest derivative and the degree of an ordinary differential equation is the power of highest derivative. Thus the differential equation, x e xy dx dy x dx y d x dx y d x = + + + 2 2 2 3 3 3 08.01.1
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08.01.2 Chapter 08.01 is of order 3 and degree 1, whereas the differential equation x dx dy x dx dy sin 1 2 2 = + + is of order 1 and degree 2. An engineer’s approach to differential equations is different from a mathematician. While, the latter is interested in the mathematical solution, an engineer should be able to interpret the result physically. So, an engineer’s approach can be divided into three phases: a) formulation of a differential equation from a given physical situation, b) solving the differential equation and evaluating the constants, using given conditions, and c) interpreting the results physically for implementation. Formulation of differential equations As discussed above, the formulation of a differential equation is based on a given physical situation. This can be illustrated by a spring-mass-damper system. Above is the schematic diagram of a spring-mass-damper system. A block is suspended freely using a spring. As most physical systems involve some kind of damping - viscous damping, dry damping, magnetic damping, etc., a damper or dashpot is attached to account for viscous damping. K b x Figure 1 Spring-mass damper system. M Let the mass of the block be M , the spring constant be K , and the damper coefficient be b . If we measure displacement from the static equilibrium position we need not consider gravitational force as it is balanced by tension in the spring at equilibrium.
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mws_gen_ode_bck_primer - Chapter 08.01 Primer for Ordinary...

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