朗斯基行列式1·.doc

朗斯基行列式 1定义:由定义在区间上的k个可微k-1次的函数所作成的行列式： 称为这些函数的朗斯基行列式。定理 1: 若函数在区间上线性相关,则它们在a,b上的朗斯基行列式.证明: 从题设可知, 存在一组不全为零的常数使得 ,依次对t微分此恒等式, 得到把方程和方程组 看成是关于的齐次线性代数方程组，那么它的系数行列式就是.由线性代数的理论我们可以知道, 要此方程存在非零解, 它的系数行列式必须为零, 即.证毕.注意:定理1的逆定理一般是不成立的. 实际上, 很容易就能给出这样的函数. 由其构成的朗斯基行列式恒为零, 但它们却是线性无关的.例1: 和 在区间上,显然有, 但它们在此区间上却是线性无关的.因为,假设存在恒等式.则当时，推得;而当时又推得.即除了之外，找不到其它不全为零的常数可以使得恒等式在区间-1,1上都成立.因此是线性无关的.推论 1: 如果向量组(函数组)在区间a,b上存在一点处的朗斯基行列式不等于零,即Wt00, 则向量组(函数组)在I上线性无关. 这个推论是定理1的逆否命题。现在我们来规定一个标准的高阶齐次线性微分方程其中都是区间上的连续函数.定理2: n阶齐次线性微分方程 一定存在n个线性无关的解.证明: 满足初值条件,的解一定存在. 因此. 根据推论1 ,这n个解一定线性无关.定理3: 如果方程的解在区间上线性无关,则 在这个区间上任何点上都不等于零,即.证明: 用反证法.假设存在一点使得.考虑关于的齐次线性代数方程组 方程组的系数行列式, 所以它有非零解.现在用这组非零常数来构造函数根据叠加原理 , x(t)也是方程的解. 从方程组, 可以知道这个解x(t)满足初值条件 x=0 也是满足初值条件的微分方程的解.由解的唯一性, 即知,即因为不全为零, 这就与线性无关的假设相矛盾.证毕.定理4(刘维尔公式):若是线性微分方程的任意n个解, 这n个解所构成的朗斯基行列式为W(t), 则W(t)满足一阶线性方程且对定义区间I上的任意有证明: 对上式两边都求对t的导数，可得 分别用乘以行列式的第一行,第二行,.,第n-1行元素，再将它们分别加到最后一行上去. 得到的行列式的最后一行的元素为 由于是微分方程的解, 代入可得 根据行列式的性质有 即 解出来就是Wt=ce-t0ta1(t)dt. 当时, . 因此WtWt0e-t0ta1(s)ds Wronskian Determinant 1Definition:The determinant which is formed by k differentiable k-1 order functions defined on the interval and their derivatives is called Wronskian determinant of these functions.Theorem 1: If functions are linearly dependent on interval , their Wronskian determinant on a,b.Proof: From the assumption, we know that there is a set of constants that are not all zero which bring ,Take derivatives of t on this identical equation in proper order , we obtainTake equation and equation set as homogeneous linear algebraic equations about , so its coefficient determinant is .From the theory of linear algebra, we know that for the existense of nontrival solution of this linear system , its coefficient determinant must be zero, i.e. The theorem has been proved.Hint:The inverse theorem of the theorem 1 generally does not hold. In fact, it is easy to give such a function group. The Wronskian determinants of this kind of function groups are always zero, but they are linearly independent.Example 1: and On interval ,it is obvious that , but they are linearly independent on the interval. Assume that there is identity . It is easy to get , so are linearly independent.Corollary 1: If there exist one point which makes the Wronskian determinant of functions is not equal to zero at this point , i.e. Wt00, the functions are linearly independent on the interval. This corollary is the converse negative proposition of Theorem 1.Now we specify a standard higher order homogeneous linear differential equationWhere are all continuous functions on interval .Theorem 2: n-order homogeneous linear differential equation must have n linearly independent solutions.Proof: The solutions satisfied the initial condition,Must exist. So . According to corollary 1 , these n solutions must be linearly independent.Theorem 3: If the solutions of equation are linearly independent on the interval , is not equal to zero at any point on the interval , i.e. .Proof: We can use reverse proving. Assume that there exist one point which makes .Consider the homogeneous linear algebraic equations about The coefficient determinant of this equation system , so it has nontrival solution .Now construct a function with this set of constantsAccording to the Principle of superposition , x(t) is also a solution of the equation. From equation system, we know solution x(t) Satisfies the initial value condition x=0 is also a solution of D.E. which Satisfies the initial value condition.From the uniqueness of solution , we obtain ,i.e.Since are not all zero , this contradicts that are linearly independent.The theorem has been proved.Theorem 4(Liouville Formula):Ifare n arbitrarily solutions of linear differential equation, and W(t) is their Wronskian determinant, then W(t) satisfy the order-1 linear equation and arbitrarily satisfy the equationProof: Take derivative on the both sides of the equation for t , we obtain Multiply the first line, second line , . , Line n-1 of the determinant by respectively, th