《高数双语》课件section 6.3.pptx
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1、Section 7.312Differential Equations of Second Order Solvable by Reduced Order MethodsIn this part,we will introduce some methods for solving some kind of second order equations of the form(,)yf x y yThere are three types of equation of this form which can be solved by reduction of order methods.We w
2、ill discuss them in the following.(1)()yf xSolving this type of equations is very simple.We need only integrate f(x)successively twice.Example 1 Find the general solution of the equation cos.yxSolutionand then12(sin)yxC dxC 12cos.xC xC 1sinyxC SinceFinish.3Differential Equations of Second Order Solv
3、able by Reduced Order Methods(2)(,)yf x yThe character of the equation is that the function f does not contain the unknown function y.dpydxWe make the transformation yp,so thatThus the equation is transformed into(,).dpf x pdx This is a differential equation of first order.If we can find its general
4、 solution denoted by1(,),pg x C then since,dypdx we have1(,).dyg x Cdx Integration again,we obtain the desired general solution12(,).yg x C dxC 4Differential Equations of Second Order Solvable by Reduced Order MethodsExample 2 Find the particular solution of the equation2(1)2xyxywith the initial con
5、ditions 01xy 03.xy andSolutionLet;ypthen.dpydxSubstituting into the equationwe have2(1)2dpxxpdxSeparating variables we obtain22.1dpxdxpx Then,the general solution is21lnln(1)lnpxC21(1).pCx5Differential Equations of Second Order Solvable by Reduced Order MethodsSolution(continued)orIntegration again
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