电脑在工程数学上之应用.pdf

NTU Nano BioMEMS Group 電腦在工程數學上之應用 Symbolic Computation in Engineering Mathematics 林致廷林致廷 Chapter 4 Ordinary Differential Equations NTU Nano BioMEMS Group 2 Outline Command List of ODE in Maxima First Order ODE Linear Second Order ODE Higher Order ODE NTU Nano BioMEMS Group 3 Command List diff expr x Give the derivative or differential of expr with respect to some or all variables in expr ode2 eqn dvar ivar Solve an ordinary differential equation of first or second order An ODE is given by eqn the dependent variable dvar and the independent variable ivar ic1 solution xval yval Solve initial value problem for first order differential equation solution is a general solution obtained by ode2 xval gives an initial value for the independent variable in the form x x0 yval gives the initial value for the dependent varibale in the form y y0 NTU Nano BioMEMS Group 4 Command List ic2 solution xval yval dval Solve initial value problem for second order differential equation solution is a general solution obtained by ode2 xval gives an initial value for the independent variable in the form x x0 yval gives the initial value for the dependent varibale in the form y y0 dval gives the initial value for the first derivative of the dependent variable with respect to independent variable in the form diff y x dy0 bc2 solution xval1 yval1 xval2 yval2 Solve boundary value problem for a second order differential equation solution is a general solution obtained by ode2 xval1 x x1 yval1 y y1 xval2 x x2 yval2 y y2 NTU Nano BioMEMS Group 5 Command List desolve eqn x desolve eqn 1 eqn n x 1 x n Solve systems of linear ordinary differential equations eqn are differential equations in the dependent variables x l x n depends f 1 x 1 f n x n Declares functional dependencies among variables for the purpose of computing derivatives If depends f x is declared diff f x yields a symbolic derivative Each argument f 1 x 1 can be the name of a variable or array or a list of names NTU Nano BioMEMS Group 6 Differential Equations Differential equation It is an equation involving ordinary or partial derivatives of a function The order of a differential equation is the order of the highest derivative occurring in the equation A first order ordinary differential equation It involves one dependent and one independent variable and a first derivative Examples The term ordinary simply means that no partial derivatives are present 0 yyxF xyy xxyy 4 2 NTU Nano BioMEMS Group 7 Solutions of Differential Equations A solution of a differential equation is any function satisfying the differential equation General solution the solution which contains one arbitrary constant Particular solution the solution obtained by making a specific choice of the constant in the general solution Initial value problem In some cases the solution of a first order differential equation subject to the condition at a given point x Ceyyy 2 02 x eyyy 44 00 0 yxyyyxF NTU Nano BioMEMS Group 8 Separable Equations A differential equation can be called as separable if it can be shown as follows Integrate both sides to obtain the solution Command diff expr x Give the derivative or differential of expr with respect to some or all variables in expr 0 xAyyByyxF dxxAdyyB NTU Nano BioMEMS Group 9 Separable Equations Example Solve the differential equation Sol 23 8yx dx dy Cx yCx y dxxdy y dxxdy y 4 4 3 2 3 2 2 1 2 1 8 1 8 1 NTU Nano BioMEMS Group 10 Homogeneous Differential Equations Since most first order differential equations are not separable The change of variables can be used to transform the one which is not separable into the one which is separable A first order differential equation is called homogeneous with the form Then it can be transformed into a separable equation by the change of variables x y f dx dy dx x du uuf dx du xuuf x y f dx dy dx du xu dx dy x y u 11 NTU Nano BioMEMS Group 11 Homogeneous Differential Equations Example Solve the differential equation Sol 323 2yyx dx dy x Cx x y Cx u dx x du u uu dx du xu xyu x y x y dx dy ln2 ln 1 4 1 11 2 1 2 Let 2 2 3 3 3 3 NTU Nano BioMEMS Group 12 Exact Differential Equations Consider a first order differential equation of the form If we can find F x y whose total differential dF is equal to M x y dx N x y dy F is a potential function of this differential equation dy dx M x y N x y is exact Test for exactness Suppose M N M y N x are continuous over R The differential equation is exact if and only if 0 dyyxNdxyxM yxN yxM dx dy yxf dx dy x N y M NTU Nano BioMEMS Group 13 Exact Differential Equations Example The total differential of a function is given by Determine u x y Sol The coefficient of dx and dy are ux and uy Integrating the first equation Therefore uycan be found as As the result dyyxdxxyxdu223 3 yxuxyxu yx 2 and 63 32 yCxyxyxu 23 3 yxyCxyCxuy2 333 cyxyxyxu 223 3 NTU Nano BioMEMS Group 14 Integrating Factors and the Bernoulli Differential Equation If the first order differential equation is not exact find a nonzero function m x y such that multiplying by mresults in an exact equation Such a function mis called an integrating factor To find an integrating factor Example 0 0 dyyxNyxdxyxMyx dyyxNdxyxM mm x N x N y M y MN x M y m m m m mm exact is 0636 Multiply 63 and 6 0636 2223 22 22 dyyxxydxxyy yyx xxyyxNxyyyxM dyxxydxxyy m NTU Nano BioMEMS Group 15 Integrating Factors and the Bernoulli Differential Equation Here are two special cases Find an integrating factor which is a function of just x or just y Find an integrating factor by simple combinations of functions y M x N Mdy d dy d M y M x N x N y M Ndx d dx d N x N y M 11 11 m m m mm m m m mm ba yxyx m NTU Nano BioMEMS Group 16 Integrating Factors and the Bernoulli Differential Equation Example Consider Sol 032 63 222 dyyxdxyxyyx 2 3 1 1 y oft independen is xfactor gintegratinan is thereThus y oft independen isit 1 1 Note exactnot isit zero not 3663 exact siit ifcheck 22 22 xx x eyyexyxF ex dx d x N y M N yxxyxx x N y M m m m m NTU Nano BioMEMS Group 17 The Bernoulli Differential Equation Bernoulli equation a is a constant 0 dyxPdxyxRyxQ yxRyxQ dx dy xP a a dx xP xQ xPxf xP xQ xP xP xf xf yxQxfyxPxfyxPxf b ybxfxRybxQxfyxPxfyxPxf yxfyx x xPxP y yxRyxQyxRxQ x xPxP xy yxRyxQyxRyxQ y x N x N y M y M bbbb b a a a a a m m m m am m m m m m m m m aaa a aa aa 1lnln 1 1 Choose 1 Let 1 1 NTU Nano BioMEMS Group 18 The Bernoulli Differential Equation Example Solve 4 yy dx dy Ceye eyeyxF dyeydxeydydxyy eye y yx xRxQxP xx xx xx x dx 3 3 implicitly defined issolution general The 3 3 01 0 1 4 1ith equation w Bernoulli a is This 333 333 34334 34 3 4 m a NTU Nano BioMEMS Group 19 Linear Differential Equation A first order linear differential equation It can be treated as a Bernoulli equation with a 0 and P x 1 To solve it Multiply by xqyxpy dxxp e dxxpdxxpdxxp dxxpdxxp dxxpdxxp dxxpdxxpdxxp Ceexqexy Cexqye exqye dx d exqyexpey NTU Nano BioMEMS Group 20 Linear Differential Equation Example Solve Solution xyysin x xxx xx xxx Cexxy Cexxdxxeye xeye xeyeey xdxdxxp cossin 2 1 cossin 2 1 sin sin sin NTU Nano BioMEMS Group 21 The Riccati Equation The Riccati Equation A Riccati equation is linear if P x 0 but is nonlinear otherwise To solve it Given one solution S x change variables xRyxQyxPy 2 PzQPSz z xQ z xSxP z xPz z xRxSxQxSsPxS z xQ z xSxP z xP xRxSxQxSxPz z xS xR z xSxQ z xSxPz z xS z z xSy z xSy 2 11 2 1 1 11 2 1 1 11 1 1 1 22 2 2 2 2 2 2 2 NTU Nano BioMEMS Group 22 The Riccati Equation Example Sol x y x y x y 211 2 x z x z xzxzx z z zzyzy 13 21 1 11 1 1 1 1 11let 2 2 2 3 3 3 3 23223 3 3 2 3 1 1 1 1 1 3 1 3 z ofequation aldifferentilinear isIt xK Kx y x C z y x C zxzxxzxzx xe dxx NTU Nano BioMEMS Group 23 Initial Value Problem Consider Sol the general solution of the linear differential equation 6y 0 1 yy x x ey KKey y Key yy 0 0 51 6110 60 problem valueinitial the 1 1 NTU Nano BioMEMS Group 24 Linear Second Order Differential Equations A second order linear differential equation The functions p q and f are called coefficient functions f is also called a forcing function For example There are other nonlinear second order differential equation xfyxqyxpy x x exfxqxxp eyyxy 2 2 4 3 2 432 x ey xyyyy 2 06 NTU Nano BioMEMS Group 25 Linear Second Order Differential Equations To obtain the full solution We will require two additional information to assign unique values This leads us to define a linear second order initial value problem For a linear second order differential equation It is homogeneous if f x 0 For example BxyAxyxfyxqyxpy 00 xfyxqyxpy 0 yxqyxpy 0in shomogeneou is 0ln 1 line real wholeon the shomogeneou is 0 yxy x y xyy NTU Nano BioMEMS Group 26 Linear Second Order Differential Equations If y1and y2be solutions of the homogeneous linear equation on the interval I y1 y2is a solution on I For any constant c c y1is also a solution on I Wronskian Test for linear independence Let y1and y2be solutions of the homogeneous linear equation on the interval a b Let Either W y1 y2 x 0 or W y1 y2 x 0 y1and y2are linearly independent if and only if W y1 y2 x 0 The W y1 y2 is called the Wronskian of y1and y2 0 yxqyxpy xyxy xyxy xyyW 21 21 21 xyxyxyxyxyyW 122121 NTU Nano BioMEMS Group 27 Linear Second Order Differential Equations Let y1and y2be linearly independent solutions of a linear second order differential equation Every soluti