4.1几类可降阶的高阶微分方程（4.1 higher-order differential equations with lower order）

4.1几类可降阶的高阶微分方程（4.1 higher-order differential equations with lower order） 4.1几类可降阶的高阶微分方程(4.1 higher-order differential equations with lower order) The first The first the first Chapter 4 chapter Differential equations and differential equations Differential equations and differential equations with differential equations The differential equation is on the first page of the difference equation Page 1 Go to the next page The next page is on the next page The next page returns home Return to the home page Return to the home page School of mathematics and computing, xiangtan university 1 the first The first the first Chapter 4 chapter Differential equations and differential equations Differential equations and differential equations with differential equations Differential equations and difference equations are in many practical problems, such as science and technology and economic management Many practical problems, such as science and technology and economic management, are in many practical problems, such as scientific and technological and economic management In many practical problems, such as science and technology and economic management, ,, , The variables in the system can often be represented as one The variables in the system can often be expressed as the variables in a system can often be represented as one The variables in the system can often be represented as one ( (( (group way Group) )) ) differential equation Differential equations of differential equations The differential equation or Or or Or difference equation Difference equation The difference equation, ,, They are two different kinds of equations They're two different kinds of equations and they're two different kinds of equations They're two different kinds of equations, ,, , the former handled the quantity The former handles the amount of the former The amount that the former handles dynamic Dynamic dynamic Dynamic page one Page 1 Go to the next page The next page is on the next page The next page returns home Return to the home page Return to the home page School of mathematics and computing, xiangtan university 2 discrete variables Discrete variables of discrete variables Discrete variables, ,, , The interval period is counted as a statistic The interval time period is counted as the interval period of statistics Interval time intervals as statistics. It's a continuous variable The continuous variable is a continuous variable It's a continuous variable; ;; ; The latter takes the non-negative integer value in turn The latter takes the non-negative integer value in sequence and the latter takes the non-negative integer value in turn The latter takes the non-negative integer value in turn For example, there is a lot of data in economic variables For example, in the economic variables there are a lot of data that are in the economic variables For example, in the data of economic variables, there are many higher order differential equations with lower order Some of the lower order differential equations of higher order differential equations Several high order differential equations with lower order a One by one A, ,,, The differential equations of type Differential equations of type differential equations Differential equations of type The second . Second, ,,, The differential equations of type Differential equation of type differential equation The first page of the differential equation of the type Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university Three of four The unevenness Four, ,,, And summary Summary summary summary The second . Second, ,,, The differential equations of type Differential equation of type differential equation Differential equations of type three Three three Three, ,,, The differential equations of type Differential equation of type differential equation The differential equations of the second and second order of the differential equations are collectively referred to as The second and second order of differential equations are collectively referred to as the second and second order of the differential equations The second and second order differential equations are collectively referred to as the higher order Advanced high order Higher order differential equations Differential equations of differential equations Differential equations. There are some higher-order differential equations Some higher-order differential equations have higher order differential equations Some higher-order differential equations, ,, You can use either the independent variable or the unknown function You can either use the independent variable or unknown function through the independent variable or unknown function You can use the independent variable or unknown function Of the the substitution Substitution substitution Change the order number Lower order Numbers Lower order number, ,, So we can figure out the solution So we can figure out what the solution is So let's figure out the solution. .. The previous page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university The solution of three kinds of high order differential equations is introduced In this paper, we introduce the solution of three kinds of higher order differential equations with reduced order In this paper, we introduce the solution of three kinds of higher order differential equations. .. A. One by one A, ,,, , () () nyfx = = = = make To make To (1), nzy? ?? ? = = = = so So so So (1) dzfxxC = + = = + + = + ? ? ? ?, ,, , i.e., Namely namely namely Differential equations of type Differential equation of type differential equation Differential equations of type Variable substitution Variable substitution Variable substitution the Then, The previous page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 5 the Namely namely namely The same can be And that's the same thing The same can be [ [[ [] ]] (2) 2 dnyxC? ?? ? = + = = + + = + ? ? ? ? [ [[ [] ]] ] dx = = = = ? ? ? The integral of Pass through in turn And that's going to be n times Integration by integration Time points, ,, , including It's got to contain it You have to have n arbitrary constants An arbitrary constant of the general solution of an arbitrary constant A general solution to any constant. .. . 12. CxC++ + + + + + + example Cases of cases Case 1 solution solving Solution ( (( (a) )) 2) 1 cosxyexdxc ' ' ' ' ' ' ' ' ' ' ' ' =? + =? + =? + =? + ? ? ? ? 2 11 Sin, 2 xexC ' ' ' ' =? + =? + =? + =? + 21 (sin) xyexCdx ' ' ' ' ' ' ' ' =? + =? + =? + =? + ? ? ? 2 cos. ? xyex ' ' ' ' ' ' ' ' ' ' ' ' =? =? =? =? Solve the previous page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 61 (sin) 2 yexCdx ' ' ' ' ' ' ' ' =? + =? + =? + =? + ? ? ? ? 21 8 Xye = = = = sinx + + + + 2 1 cx + + + + 23 CXC + + + + + + + + cosx + + + + 12, CxC ' ' ' ' + + + + + + + + ( (( (here Here here Here the 21st 4 xe = = = 12 + =. + + + = = = = ' ' ' ' ' ' ' ' ' ' ' ' xy31 , 3 yxxC ' ' ' ' ' ' ' ' = + + = + + + + = + + 42 211 , 122 yxxCxC ' ' ' ' = + + + + + + + + + = = = + + + example Cases of cases Example 2 solve differential equations Solve differential equations with differential equations Differential equations. Solutions solving We can integrate both sides of this equation The integral of both sides of the equation is the integral of both sides of the equation Integrate both sides of the equation: : : : I'm going to integrate it with the second order equation I can integrate the second order equation with respect to the second order equation I can integrate the second order equation Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 7122532 23111 6062 YxxCxCxC = + + + + + + + + + + + + = = This is equal to + + + + and finally the integral of the first order equation Finally, the integral of the above first-order equation is the integral of the above first-order equation Finally, the integral of the first order equation, ,, It has to be solved The general solution is to have a general solution The general solution is 532 12311 . 606 XxCxCxC = + + + + + + + + + + + + = = = + + + + (,) yfxy ' ' ' ' ' ' ' ' ' ' ' ' = = = Is equal to the differential equation Differential equation of type differential equation Differential equations of type set Set a set (), ypx ' ' ' ' = = = The original equation is the first order equation The original equation is reduced to a first order equation The original equation becomes a first order equation Let's call it the general solution The general solution is set to its general solution Let's say that the general solution is 1 (,), PxC? ?? ? = = = = would have to Is much better to get the Would have to Second, the Then, the Variable substitution Variable substitution The variable is changed to one page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 81 (,). YxC? ?? ? ' ' ' ' = = = = reintegration Once again, the integral is once again So once again, ,, The general solution to the original equation The general solution of the original equation The general solution to the original equation 12 (,) d. YxCxC? ?? ? = + = = + + = + ? ? ? ? cases Cases of cases Example 3 to solve Solution to solve Solving (1) 2, xyxy ' ' ' ' ' ' ' ' ' ' ' ' + = + + = = + = 01, y = = = = = = = = 0, 3), y = = = = ' ' ' ' = = = = solution solving Solution to To make make Generation into the equation Substitute into this equation Plug in the equation, ,, , Have to get 2 (1) 2 XPXP ' ' ' ' + = + + = = + = separated variables Separation of variables The separation of variables is integral I have to integrate it Integral to 2 one LNLN (1) ln, pxC = + + = + + + + = + +, Then, The previous page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 910 3, x, y = = = = ' ' ' ' = = = = use use The use of 13, C = = = = have to Have to get So have And so there is So there are 2 3 (1). Yx ' ' ' ' = + = = + + It's equal to plus both ends I can integrate both ends You get 3 at both ends 2 3. YxxC = + + = + + + + = + + using use using 01, y = = = = = = = = 21, C = = = = have to Have to get 331. Yxx = + + = + + + + So it's going to be equal to plus plus So the particular solution is therefore the particular solution So the particular solution is three Three three Three, ,,, , (,) yfyy ' ' ' ' ' ' ' ' ' ' ' ' = = = Is equal to the differential equation Differential equation of type differential equation Differential equations of type make To make Make), (ypy = = = = ' ' ' ' d d p y x ' ' ' ' ' ' ' ' = = = = dd dd py yx =? =? =? =? equation Therefore, the equation is transformed into an equation equation Let's call it the general solution The general solution is set to its general solution Let's call it the general solution For a quick A quick quick For a quick Variable substitution Variable substitution Variable substitution the Then, (,), pyC ? ?? ? = = = = back Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university Let's say that this is the general solution The general solution is set to its general solution Let's call it the general solution For a quick A quick quick For a quick The integral of the separation variable After separating variables, the integral separates the variables The integral of the separation variable, ,, The general solution to the original equation The general solution of the original equation The general solution of the original equation 1 (,), pyC ? ?? ? = = = It's equal to the input equation Substitute the input equation into the equation Substitution equation Both ends integrate The two ends integrate both ends Both ends integrate LNLNLN, pyC = + = = + + PCy = = +, = = = that is Namely namely The solution solving Solution set Set a A d d p y x ' ' ' ' ' ' ' ' = = = = dd dd py yx = = = = d . d p p y = = = =, Then, the case Cases of cases Example 4 to solve Solution to solve To solve the 2 = 0) ( = = = ' ' ' ' ? ?? ? ' ' ' ' ' ' ' ' Yyy back Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 11 points The two ends integrate both ends So it's 1 at both ends LNLNLN, pyC = + = = + + = + 1, pCy = = = = that is Namely namely That is, (( The first order linear homogeneous equation The first order linear homogeneous equation First order linear homogeneous equation )) ) So we have the general solution Therefore, the general solution is the general solution So this is the solution solving Solution to To make Make? ?? ? ? ?? ? ? ?? ? 20, yye ' ' ' ' ' ' ' ' ? = ? =? = ? = 00, y = = = = = = = = 01. Y = = = = ' ' ' ' = = = = (), ypy ' ' ' ' = = = = d , d p yp y ' ' ' ' ' ' ' ' = = = = substituted into equation Substitute into this equation Plug in the equation, ,, , Have to get Have to, Then, the case Cases of cases Example 5 solve the initial value problem Solve the initial value problem The initial value problem is one page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 12 points to I have to integrate it Integral to Initial condition Use the initial conditions to use the initial conditions Using the initial conditions, ,, Ten, C = = = = have to Have to get 22 111 . 22 ypec = + = = + + = + 2, yexC? ?? ?? = + ? = +? = + ? = + points to I have to integrate it Integral to d . d yy PE x = = = = = = = = have to Have to get Have to 00 10, yxpy = = = = = = = = ' ' ' ' = = > = = > = = > = = > according to According to according to According to the previous page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 13 1. Yex? ?? ? ? = ? =? = ? = 21. C =? =? =? =? Have to Have to get 00, y = = = = = = = = again And then again Again by by So let's just solve for this Therefore, the particular solution is the particular solution So this is 2, 4 The unevenness Four, ,,, And summary Summary summary summary The solution of the differential equation with reduced order The solution of the differential equation of the differential equation can be reduced by decreasing order differential equation The method of reducing order differential equation - descending order The descending order method Order reduction method Successive integral I'm going to integrate this one by one It's one page at a time Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 14 to (), ypx ' ' ' ' = = = = make To make make (), ypy ' ' ' ' = = = =, Then, the the Then, Think and practice Think and practice thinking and practice Thinking and practice 1. The equation equation How do we solve this equation How do I solve for substitution How do you solve it? a Answer answer A: : : : To make make or Or or or The general said Generally speaking Generally said, ,, It's easier to use the former The former is more convenient to use the former The former is more convenient. .. . All can Are all can All can. .. . Sometimes it is convenient to use the latter Sometimes it is convenient to use the latter Sometimes it is convenient to use the latter. .. . For example, For example, for example, For example, ,, The previous page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 152. To solve the initial value problem of second order differential equation, we should pay attention to the problems In order to solve the initial value problem of second order differential equation, we should pay attention to the problem of the initial value of second order differential equation To solve the first order differential equation of second order differential equation, which problems should be paid attention to? a Answer answer Answer: (1) general situation General situation generally In general, ,, , it is easy to calculate the edge constant The method is simple and simple to solve the constant calculation of edge solution The boundary solution constant is simple to calculate. .. . (2) when the square is open When you square the square, you meet the square When I square it, ,, In order to determine the positive and negative Numbers According to the question, the sign should be determined according to the question Make sure the sign is positive. .. . case Cases of cases Case 5 a One by one A, ,,, To solve the general solution of each differential equation Find the general solution of the following differential equations Find the general solution to the following differential equations: : : : 1. ,,, , xxe Y = = = = ' ' ' ' ' ' ' ' ' ' ' '; ;; ; 2, ,,, Yy, 21 ' ' ' ' + + + + = = = = ' ' ' ' ' ' ' '; ;; ; 3, ,,, , y yy ' ' ' ' + + + + ' ' ' ' = = = = ' ' ' ' ' ' ' '(3) (; ;; ; 4, ,,, , 0 one 22 = = = = ' ' ' ' ? ?? ? + + + + ' ' ' ' ' ' ' ' y y Y. .. . The second . Second, ,,, To satisfy the particular solution to the initial conditions To satisfy the particular solution to the initial conditions, the following differential equations satisfy the following differential equations To satisfy the particular solution to the initial conditions of the following differential equations: : : : 1. ,,, , 0 1011 1, 3 = = = = ' ' ' ' = = = = = = = = + + + + ' ' ' ' ' ' ' '= = = = = = = = xxyyyy; ;; ; practice practice practice Blow gently Problem sets Topic title Title page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 161 , ,,, ,1,0111, 0 = = = = ' ' ' ' = = = = = = = = + + + + ' ' ' ' ' ' ' '= = = = = = = = xxyyyy; ;; ; 2, ,,, , 1, , 0000, 2? ?? ? = = = = ' ' ' ' = = = = = = = = ' ' ' ' ? ?? ? ' ' ' ' ' ' ' '= = = = = = = = xxyyyay; ;; ; 3, ,,, , 2, , 1300 = = = = ' ' ' ' = = = = = = = = ' ' ' ' ' ' ' '= = = = = = = = xxyyyy. .. . three Three three Three, ,,, Try, please Try to try for Yx trying o ' ' ' ' ' ' ' ' = = = The passing point of = The passing point of the point It goes through the point (0, 1) M and at this point and the line And at this point and the line and at this point and the line And at this point and line 1 2 x Y = + = = + + Is equal to the integral curve of tangent The integral curve of tangent integral curve The tangent integral curve . .. . Practice questions The answers to the exercise questions The answer is one One by one A, ,,, , 1, 11 1. ,,, , 32 12 3 CXCX C exeyx X + + + + + + + + + + + +? ?? ? = = = =; ;; ; 2 22 2, ,,, , 2, 1) cos (lnCCxy + + + + + + + +? ?? ? = = = =; ;; ; 3 33 3, ,,, , 1, (2) arcsin CeCyx + + + + = = = =; ;; ; 4 44 4, ,,, , x CxC y one one + + + + ? ?? ? = = = =. .. The previous page Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 17 4 44 4, ,,, , xCxC y2 11 + + + + ? ?? ? = = = =. .. . The second . Second, ,,, , 1, 11 1. ,,, , 22 Xxy? ?? ? = = = =; ;; ; 2 22 2, ,,, , 1 ln () one + + + +? ?? ? = = = = ax a. Y; ;; ; 3 33 3, ,,, And 4) one 2 one (+ + + + = = = = y. .. . three Three three Three, ,,, , 1, 2 one 6 13 + + + + + + + + = = = Combinations = xxy. .. . As a make homework Industry industry Industry back Page 1 Go to the next page The next page is on the next page The next page returns to the front page Return home page to return home page Return to the home page School of mathematics and computing, xiangtan university 18