Ch3 - Ch.3 PHYS2004 Quantitative Methods for Basic Physics...

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Ch.3 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 1 Chapter 3 Ordinary Differential Equations An ordinary differential equation is an equation containing ordinary (not partial) derivatives of the unknown function. The order of an ordinary differential equation is the order of the highest derivative that appears in the equation. Generally, the equation    ,, , , 0 n Fxyy y , (3.1) is an ordinary differential equation of the n th order. To solve the differential equation (3.1) is to find the function   yf x that satisfies it. The differential equation (3.1) is said to be linear if F is a linear function of the variables ,,, n y yy . Thus the general linear ordinary differential equation of order n is       1 10 nn axy a xy axygx  . (3.2) 3.1 First-order Differential Equations A first-order differential equation is given by: , x y   , ( 3 . 1 . 1 ) where f is a given function of two variables. 3.1.1 Direct Integration Any first-order differential equation of the form x   , ( 3 . 1 . 2 ) can be solved by direct integration and the solution is given by
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Ch.3 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 2  y fxd x . ( 3 . 1 . 3 ) 3.1.2 Separation of Variables In general, the first order differential equation (3.1.1) cannot be integrated directly because the variables x and y appear together on the R.H.S. However, if the variables x and y in , f xy can be separated:  , gxhy , ( 3 . 1 . 4 ) the equation (3.1.1) can be rewritten as 1 yg x hy   , ( 3 . 1 . 5 ) which can be solved by direct integration on both sides: 1 dy g x dx  . ( 3 . 1 . 6 ) 3.1.3 Exact Equations If the differential equation (3.1.1) can be written as: ,, 0 Mxy Nxyy , ( 3 . 1 . 7 ) or 0 Mxyd x Nxyd y  , (symmetric differential form) and there exits a function such that , , xy Mxy x ,   , , Nxy y , (3.1.8) the differential equation becomes ,0 d dx , ( 3 . 1 . 9 )
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Ch.3 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 3 and the solution is simply  , x yC . ( 3 . 1 . 1 0 ) Example Solve the differential equation 32 2 23 0 . dy xy x y dx  By inspection, it can be seen that the left-hand side is the derivative of x y . Thus the given equation can be rewritten as 0 d xy dx , and the solution is given implicitly by x and explicitly by 2/3 y kx . For more complicated equations it may not be possible to recognize that the L.H.S. of (3.1.7) is the derivative of some function   , x y by inspection. Instead, there is a systematic procedure for determining whether a given differential equation is exact by considering the partial derivatives of   , Mxy and   , Nxy (see Appendix A for details). 3.1.4 Homogeneous Equations A differential equation of the form , dy f xy dx , is said to be homogeneous whenever the function f does not depend on x and y separately, but only on their ratio yx or x y . Thus homogenous equations are of
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Ch.3 PHYS2004 Quantitative Methods for Basic Physics I (1 st Term 10/11) 4 the form dy y F dx x 
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This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.

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Ch3 - Ch.3 PHYS2004 Quantitative Methods for Basic Physics...

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