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单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,上页,下页,铃,结束,返回,首页,Chapter 4,Higher-Order Differential Equation,4.1 Preliminary Theory:Linear Equation,上页,下页,铃,结束,返回,首页,4.1.1 Initial-value and Boundary-value Problems,4.1.2 Homogeneous Equation,Chapter 4 Higher-Order Differ,For a linear differential equation,an nth-order initial-value problem is,Initial-value problem,4.1.1 Initial-value and Boundary-value Problems,For a linear differential e,Another type of problem consists of solving a linear,Boundary-value problem,differential equation of order two or greater in which,the dependent variable,y,or its derivatives are,specified at different points.A problem such as,Another type of problem consis,is called a boundary-value problem.The prescribed,values,are called boundary,conditions.,For a second-order differential equation,other pairs of boundary conditions could be,is called a boundary-value pro,Theorem 4.1,Existence of a Unique Solution,on an interval,I,and let,Let,and,g(x),be a continuous,for every,x,in this,interval.If,is any point in this interval,then a,solution,y(x),of the initial-value problem(1)exists,on the interval and is unique.,Theorem 4.1 Existence of a,4.1.2 Homogeneous Equation,with,g(x),not identically zero,is said to be,A linear nth-order differential equation of the form,Is said to be,homogeneous,whereas an equation,non-homogeneous,.,4.1.2 Homogeneous Equati,We make the following important assumptions about,the equation(2)and(3),are continuous;,(1)the coefficients,(2)The right-hand member g(x)is continuous;,(3)for every x in the interval I.,We make the following importan,Differential Operator,The symbol,D,is called a,differential operator,（微分算子）,.,We define an,nth-order differential operator,to be,Differential OperatorThe symbo,For example,and,For exampleand,As a consequence of two basic properties of,differentiation,The differential operator L possesses a linearity property,where are constants,and the nth-order,differential operator L is a linear operator.,As a consequence of two basic,Superposition principle,（叠加原理）,Theorem 4.2 Superposition principle,(homogeneous equation),Let be solutions of the homogeneous,n,th-order differential equation(6)on the interval,I,.,Then the linear combination,Superposition principle（叠加原理）,where the are arbitrary,constants,is also a solution on the interval.,Corollaries,（推论）,to Theorem 4.2,(A)a constant multiple of a,solution of a homogeneous linear,differential equation is also a solution.,(B)a homogeneous linear differential equation,always possesses the trivial solution,y=0.,where the,Example 1,The function and are both,solutions of the homogeneous linear equation,on the interval .By the,superposition principle,the linear combination,Is also a solution of the equation on the interval.,Example 1The function,
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