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
6、we obtain the general solution of the given equation,Substituting the initial conditions 03,xy 01xy into the equation(1)21(1)dyCxdx(1)21213yCxxC(2)and(2),respectively,we obtain that13C and21.C Finish.Example 2 Find the particular solution of the equation2(1)2xyxywith the initial conditions 01xy 03.x
7、y and6Differential Equations of Second Order Solvable by Reduced Order Methods(3)(,)yf y yThe character of the equation is that the function f does not contain the independent variable x.dpydxWe make the transformation yp ,so thatThus the equation is transformed into(,).dpf y pdx(3-1)If we regard p
8、as an unknown function and y as the independent variable in equation(3-1);then by the rule for differentiation of composite functions we have.dpdp dydpypdxdy dxdy7Differential Equations of Second Order Solvable by Reduced Order Methods(,)dpf y pdx(3-1)Hence the equation(3-1)may be changed into(,).dp
9、pf y pdy(3-2)If the general solution of(3-2),denoted by p=g(y,C1),can be found then since p=y we have1(,).dyg y Cdx Therefore,the general solution of the given equation is given by21.(,)dyxCg y C 8Differential Equations of Second Order Solvable by Reduced Order MethodsSolution Substituting into the
10、given equation we obtain20,dpyppdythat is,0dpp ypdyor0,0.pdpypdy From p=0,we have0,dydx so y=C.Example 3 Find the general solution of the equation 2()0.yyy.dpydxLetyp ,so that9Differential Equations of Second Order Solvable by Reduced Order MethodsSolution(continued)From the equation0dpypdywe can obtain1pC y or1,dyC ydx so that12.C xyC e It is easy to see that the solution y=C may be obtained from the general solution by choosing C1=0.Therefore,the general solution of the given equation is the one shown in the last equation.Finish.Example 3 Find the general solution of the equation 2()0.yyy
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