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
20、(5)18Nonhomogeneous Linear Differetial EquationThus,we have the general solution for the differential equation(4)where C is an arbitrary constant.Also,the integration of function P(x)does not need more general solution.()()()P x dxP x dxyeCQ x edx (7)The general solution of a non-homogeneous linear
21、equation consists of the sum of one of its particular solutions and the general solution of the corresponding homogeneous linear equation.()()()()P x dxP x dxP x dxyCeeQ x edx 19Nonhomogeneous Linear Differential EquationSolution IStep 1.Put the equation in standard form to identify P and Q.33,(),()
22、dyyxP xQ xxdxxx Example 5 Solve the equation23,0.dyxxyxdxStep 2.Find the solution of the homogeneous equation3.dyydxx 3().P x dxCxyCe Step 3.Set then find h(x).3(),yh x x 20Nonhomogeneous Linear Differential EquationSolution IExample 5 Solve the equation23,0.dyxxyxdx3223()()3(),xxh x xh xh xx 3()yh
23、x x 21(),h xx 1().Ch xx Then,the solution is 23,0.yxCxx Finish.21Nonhomogeneous Linear Differential EquationSolution IIStep 1.Put the equation in standard form to identify P and Q.33,(),()dyyxP xQ xxdxxx Step 2.Find an antiderivative of P(x)(any one will do).31()33ln3ln(0)P x dxdxdxxxxxx Step 3.Subs
24、tituting into the formula()()()()P x dxP x dxP x dxyCeeQ x edx Example 5 Solve the equation23,0.dyxxyxdx22Nonhomogeneous Linear Differential EquationSolution II(continued)()()()()P x dxP x dxP x dxyCeeQ x edx The solution is 23,0.yxCxx 3ln3ln3ln()xxxCeex edx 33331211CxxdxCxxCxx 23.xCx Finish.Example
25、 5 Solve the equation23,0.dyxxyxdx23Nonhomogeneous Linear Differential EquationExample 6 Find the general solution of the equation3dyydxyx and the particular solution which satisfies the initial condition01.xy Solution Since the equation can be rewritten as follows321dxyxxydyyy then it is a linear d
26、ifferential equation of first order with the unknown function().xx y The corresponding homogeneous equations is1dxxdyy 24Nonhomogeneous Linear Differential EquationSolution(continued)By separation of variables,its general solution isxCy To find the general solution,we let C=h(x),and we obtain(),xh x
27、 y so()()dxh y yh ydy Substituting this equation into the original equation,we have2()()()h y yh yh yy()h yyExample 6 Find the general solution of the equation3dyydxyx and the particular solution which satisfies the initial condition01.xy 25Nonhomogeneous Linear Differential EquationSolution(continu
28、ed)So that21()2h yydyCyC Thus the desired general solution is312xyCySubstituting the initial condition into the general solution,we obtain01xy 12C 21(1).2xy yFinish.Example 6 Find the general solution of the equation3dyydxyx and the particular solution which satisfies the initial condition01.xy 26Be
29、rnoullis EquationDefinition An equation of the form()()(0,1)dyP x yQ x ydx is called Bernoullis equation.Bernoulli,Jakob(1654-1705)Swiss MathematicianBernoulli,Johann(1667-1748)Swiss MathematicianBernoulli,Daniel(1700-1782)Swiss Mathematician27Bernoullis EquationTherefore,the general solution of the
30、 given Bernoullis equation may be obtained from the general solution of the last equation by the transformation .If we want to obtain the total solution,we need to check if y=0 is also a solution.1zy Bernoullis equation can be reduced to a linear equation by a kind of Transformation.To show this,div
31、iding both sides of the equation by we obtainy 1()()dyyP x yQ xdxLet1zy ,we have(1)dzdyydxdx (1)()(1)()dzP x zQ xdxThus,we obtain 28Bernoullis EquationExample 7 Find the general solution of the equation23.xdyxyeydx Solution It is easy to see that the equation is a Bernoullis equation.232.xdyyxyedx S
32、ubstituting into the last equation,Let z=y-2,so that 32.dzdyydxdx Then,its general solution is2(2)xzexC Therefore,the general solution of the given equation is22(2)xyexC()()()()P x dxP x dxP x dxyCeeQ x edx Dividing both sides of the equation by y3 we havewe obtain222.xdzzxedx 29Bernoullis EquationS
33、olution(continued)It is easy to see that y=0 is also a solution of the given equation,but it can not be obtain from the general solution by choosing C.22.20 xeyxCy Hence,the total solution isFinish.Example 7 Find the general solution of the equation23.xdyxyeydx 30Some Other Kinds of Equations That C
34、an Be Solved by Transformations of VariablesExample 8 Find the particular solution of the equation 220yyxyx satisfying the initial condition01.xy Solution This equation can be rewritten in the form:221()202yxyxthen,let y 2=u,it becomes the linear equation42uxux The general solution of the last equat
35、ion is easily found to be2212xuCe()()()()P x dxP x dxP x dxyCeeQ x edx Thus the general solution of the given equation is22212xyCe The initial condition2221(1).2xye Finish.31Some Other Kinds of Equations That Can Be Solved by Transformations of VariablesExample 9 Find the general solution of the equ
36、ation cos().yxySolution Let u=x+y,so1uy .Thus the given equation may betransformed into2cos12cos2uuu By separation of variables we have22cos2dudxu Thus,tan2uxCand the general solution may be found by the transformation u=x+y.Finish.32Some Other Kinds of Equations That Can Be Solved by Transformation
37、s of VariablesExample 10 Find the general solution of the equation(lnln).xyyyxy Solution Since the right side of the given equation may be rewritten as yln(xy),and the left side is just()dxydx,we try to make the transformationu=xy.Then the given equation becomes tolnduyudx This equation is not suita
38、ble because it contains three variables,but if we writeuyx,it may be changed intolnduuudxx Separating variables we havelndudxuux 33Some Other Kinds of Equations That Can Be Solved by Transformations of VariablesSolution(continued)Taking integration on both sides,we haveln lnlnlnln,uxCCxthat is,Cxue Hence the general solution of the given equation is1.Cxyex Finish.Example 10 Find the general solution of the equation(lnln).xyyyxy
