《高数双语》课件section 6.2.pptx
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1、Section 7.22First-Order Separable Differential Equations一阶可分离变量方程一阶可分离变量方程Definition(First-order equations with variables separable)The equation(,)yf x yis separable(可分离变量的可分离变量的)if F is the product of a function of x and a function of y.The differential equation then has the form(,)()()dyf x yg x h
2、 ydxThe general form of a first-order differential equation isF(x,y,y)=0.If this equation can be solved for y,it can be written in the formy=f(x,y).3First-Order Separable Differential EquationsExample 1 Find the general solution of the equation2dyxydx Solution It is easy to see that the equation has
3、 variables separable.separating variables we have2dyxdxy Integration on both sides gives2dyxdxy so that21ln yxC0;y Let 4First-Order Separable Differential EquationsSolution(continued)21ln yxCHence or2211xCCxyee e 2xyCe where 1CCe is an arbitrary non-zero constant.It is obvious that y=0 is also a sol
4、ution of the given equation,and we can see that it may be included in the general solution if the constant C is allowed to be zero.Therefore,the general solution of the given equation is ,when C is an arbitrary constant and this is also the total solution of the equation.2xyCe Finish.Example 1 Find
5、the general solution of the equation2dyxydx 5First-Order Separable Differential EquationsExample 2 Solve the differential equation2(1)xdyy edxSolutionSince 1+y2 is never zero,we can solve the equationby separating the variables.22221(1)(1)11tanxxxxxdyy edxdyy e dxdye dxydye dxyyeC Treat dy/dx as a q
6、uotient of differentials and multiply both sides by dxIntegrate both sides.Divide by(1+y2).C represents the combined constantsof integration.6First-Order Separable Differential EquationsSolution(continued)The equation 1tanxyeC gives y as animplicit function of x.,we can solve for y as an explicitfun
7、ction of x by taking the tangent of both sides:1tan(tan)tan()xyeC/2/2xeCWhentan()xyeCFinish.Example 2 Solve the differential equation2(1)xdyy edx7Homogeneous Equation(齐次微分方程齐次微分方程)Definition(homogeneous equation)If an equation is of the form:dyyfdxx where f is any continuous function,then it is call
8、ed a homogeneous equation(齐次微分方程齐次微分方程).For instance,22dyydxx and31dyydxxhomogeneous equationsSince,equation323dyxydxxy 3213yxyx it is also homogeneous equation.8 Homogeneous EquationTo solve the homogeneous equation,let ,or ,soyux yux dyduuxdxdxThen the differential equation becomes todyyfdxx()duux
9、f udxor()duxf uudxThis is a separable equation.If we find the solution u=u(x,C),the general solution of the original equation can be obtained by the substitution yux(,).yxu x C or9 Homogeneous EquationExample 3 Find the general solution of the equation2.dyxyxydxSolution Dividing by x on both sides o
10、f the equation andtransposing terms we have2dyyydxxxThis is a homogeneous equation.Let ,or ,thenyux yux.dyduuxdxdxSubstituting into the equation,we have2.duxudx 10 Homogeneous EquationSolution(continued)Solving by separation of variables we have(0).2dudxuxuIntegrating on both sides,1lnuxCorueCx By t
11、he transformationyux,we obtain the general solution of the given equationyxeCx or2(ln)yxCx It is easy to see that u=0 also satisfies the equation.Example 3 Find the general solution of the equation2.dyxyxydx11Homogeneous EquationSolution(continued)2(ln)0yxCxy Note that u=0 is equivalent to y=0;thus
12、y=0 is also a solution of the given equation.But it is easy to see that y=0 is not contained in the general solution(it can not be obtained from the general solution by choosing the constant C).Therefore the total solution of the given equation isFinish.Example 3 Find the general solution of the equ
13、ation2.dyxyxydx12Homogeneous EquationSolution1,duuyudy Therefore,Example 4 Find the general solution of the equation.yyxdydx 1 or1x ydydx 1.x ydxdy i.e.This is a homogeneous equation.Let ,or ,thenxuy xyu.12dyyduu 212.uCy By the transformationxuy,we obtain the general solution of the given equation22
14、12 or 2.xCyyxyCy Finish.13Linear First-Order Differential EquationsDefinition(Linear First-Order Differential Equation)A first-orderdifferential equation that can be written in the formwhere P and Q are functions of x,are a linear first-order equation(一一阶线性微分方程阶线性微分方程)and this equation is the equati
15、ons standard form.()()dyP x yQ xdxor()()yP x yQ x(1)()0dyP x ydxor()0yP x yIf Q(x)0,then equations become to which are called a homogeneous linear differential equation(齐次齐次线性微分方程线性微分方程),and if Q(x)is not always zero,the corresponding equations are called a nonhomogeneous linear differential equatio
16、n(非齐次线性微分方程非齐次线性微分方程).(2)14Homogeneous Linear differential EquationIt is easy to see that the homogeneous linear equation()0dyP x ydxmay be solved by separation of variables as follows:()dyP x dxy Integration of both sides gives1ln()yP x dxC so()P x dxyCe where C is an arbitrary constant.This is the
17、 general solution of the homogeneous linear differential equation.(3)15Nonhomogeneous Linear Differential EquationWe solve the equation()()dyP x yQ xdx(4)How to solve the non-homogeneous linear differential equation?by method of variation of constants(常数变易法常数变易法).16Nonhomogeneous Linear Differential
18、 Equation()()dyP x yQ xdx(4)Hence,to find the solution,we need only determine the function h(x).Now()()()()()P x dxP x dxdydh xeP x h x edxdx(6)Then must be a function of x,denoted by h(x).Thus the desired solution has the form:()()P x dxy xe ()()P x dxyh x e (5)Suppose that y=y(x)is a solution of t
19、he equation(4).17Nonhomogeneous Linear Differential EquationSubstituting(5)and(6)into the equation(4),we have()()()()()()()()()P x dxP x dxP x dxh x eP x h x eP x h x eQ xor()()()P x dxh xQ x e()()()P x dxh xQ x edxC ()()dyP x y Q xdx (4)()()()()()P x dxP x dxdydh xeP x h x edxdx(6)()()P x dxyh x e
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