1、上页下页铃结束返回首页 Chapter 4 Higher-Order Differential Equation4.1 Preliminary Theory:Linear Equation上页下页铃结束返回首页4.1.2 Homogeneous Equation上页下页铃结束返回首页)(),.,(),(21xyxyxyn0),.,(21nyyyW0),.,(21nyyyWdependentlinearly上页下页铃结束返回首页nccc,.,21bxatyctyctycnn,0)(.)()(221111221122(1)(1)(1)1122()().()0()().()0.()().()0nnn
2、nnnnnnc y xc yxc yxc y xc yxc yxc yxc yxc yx上页下页铃结束返回首页nccc,.,21),.,(21nyyyW)(0),.,(21bxayyyWn上页下页铃结束返回首页0),.,(21nyyyW0),.,(21nyyyW10001)(21ttttx10010)(22ttttx上页下页铃结束返回首页11t0)(),(21txtxW0)()(2211txctxc0011ct0102ct01121cct)(),(21txtx上页下页铃结束返回首页nyyy,.,21),.,(21nyyyW上页下页铃结束返回首页nyyy,.,21上页下页铃结束返回首页nyyy,
3、.,21)8()(.)()(2211xycxycxycynnnici,.,2,1,上页下页铃结束返回首页linear independence12(),(),.,()0nW y xyxyx1212(1)(1)(1)120nnnnnnyyycccyyycccyyyccc上页下页铃结束返回首页12tan,.,nthe consts c cc areindependent each other(8)(6)is general solution of上页下页铃结束返回首页Example 1The functionsxey31andxey32are bothsolutions of the homoge
4、neous linear equation09 yyon the interval),(.By inspection,the solutions are linearly independent on the x-axis.This fact can be corroborated by observing that theWronskian0633),(333333xxxxxxeeeeeeW上页下页铃结束返回首页for every x.we conclude that1yand2yform afundamental set of solutions and,consequentlyxxececy3231is the general solution of the equation on the interval.
