Sections of the site
Editor's Choice:
- Menu templates in Word: download and print
- Classroom teacher at school: functions, responsibilities, work system
- Simple math of Bayes' theorem
- Pearson correlation criterion Correlation coefficient 1 means
- Fuzzy and random sets Basic equivalences of propositional algebra
- Heading: Corporate identity
- Probability theory random events
- Small-sample statistics
- Okwed for training without a license
- Probabilistic-statistical methods of decision-making Estimation of distribution of quantity
Advertising
Solve the differential equation by the constant variation method. ODE. Variation method of an arbitrary constant. Social transformations. State and Church |
Variation method of arbitrary constants The method of variation of arbitrary constants for constructing a solution to a linear inhomogeneous differential equationa n (t)z (n) (t) + a n − 1 (t)z (n − 1) (t) + ... + a 1 (t)z"(t) + a 0 (t)z(t) = f(t) consists in replacing arbitrary constants c k in general solution z(t) = c 1 z 1 (t) + c 2 z 2 (t) + ... + c n z n (t) corresponding homogeneous equation a n (t)z (n) (t) + a n − 1 (t)z (n − 1) (t) + ... + a 1 (t)z"(t) + a 0 (t)z(t) = 0 to auxiliary functions c k (t) whose derivatives satisfy the linear algebraic system The determinant of system (1) is the Wronskian of the functions z 1 ,z 2 ,...,z n , which ensures its unique decidability with respect to. If are antiderivatives for, taken at fixed values of the integration constants, then the function is a solution to the original linear inhomogeneous differential equation. Integration inhomogeneous equation in the presence of a general solution to the corresponding homogeneous equation, it is thus reduced to quadratures. Variation method for arbitrary constants for constructing solutions of a system of linear differential equations in vector normal formconsists in constructing a particular solution (1) in the form where Z(t) is the basis of solutions of the corresponding homogeneous equation, written in the form of a matrix, and the vector function that replaced the vector of arbitrary constants is defined by the relation. The desired particular solution (with zero initial values at t = t 0 has the form For a system with constant coefficients, the last expression is simplified: Matrix Z(t)Z- 1 (τ) called the Cauchy matrix operator L = A(t) . A method for solving linear inhomogeneous differential equations of higher orders with constant coefficients by the method of variation of the Lagrange constants is considered. The Lagrange method is also applicable to solving any linear inhomogeneous equations if the fundamental system of solutions of the homogeneous equation is known. ContentSee also: Lagrange method (variation of constants)Consider a linear inhomogeneous differential equation with constant coefficients of an arbitrary n-th order: The solution is carried out in two stages. In the first step, we discard the right-hand side and solve the homogeneous equation. As a result, we get a solution containing n arbitrary constants. At the second stage, we vary the constants. That is, we consider that these constants are functions of the independent variable x and find the form of these functions. Although here we consider equations with constant coefficients, but Lagrange's method is also applicable to solving any linear inhomogeneous equations... For this, however, the fundamental system of solutions of the homogeneous equation must be known. Step 1. Solving the homogeneous equationAs in the case of first-order equations, we first look for a general solution to the homogeneous equation, equating the inhomogeneous right-hand side to zero: Step 2. Variation of constants - replacement of constants with functionsIn the second step, we will tackle the variation of constants. In other words, we will replace constants with functions of the independent variable x: If we substitute (4) into (1), then we get one differential equation for n functions. Moreover, we can relate these functions by additional equations. Then you get n equations, from which you can determine n functions. Additional equations can be constructed in various ways. But we will do it so that the solution has the most simple form. For this, during differentiation, it is necessary to equate to zero the terms containing derivatives of functions. Let's demonstrate this. To substitute the proposed solution (4) into the original equation (1), we need to find the derivatives of the first n orders of the function written in the form (4). We differentiate (4) by applying the rules for differentiating the sum and product: Find the second derivative in the same way: Thus, if you choose the following additional equations for the functions: Find the nth derivative: Substitute into the original equation (1): As a result, we got the system linear equations for derivatives: Solving this system, we find expressions for the derivatives as a function of x. Integrating, we get: Note that we have nowhere used the fact that the coefficients a i are constant to determine the values of the derivatives. That's why Lagrange's method is applicable to solving any linear inhomogeneous equations if the fundamental system of solutions to the homogeneous equation (2) is known. Examples ofSolve the equations by the method of variation of constants (Lagrange). Solving higher-order equations by the Bernoulli method Solution of higher order linear inhomogeneous differential equations with constant coefficients by linear substitution The method of variation of arbitrary constants is used to solve inhomogeneous differential equations. This lesson is intended for those students who are already more or less well versed in the topic. If you are just starting to get acquainted with DU, i.e. If you are a teapot, I recommend starting with the first lesson: Differential equations of the first order. Examples of solutions... And if you are already finishing, please discard the possible preconceived notion that the method is difficult. Because it's simple. In what cases is the method of variation of arbitrary constants applied? 1) The method of variation of an arbitrary constant can be used to solve linear non-uniform DE of the 1st order... Since the equation is of the first order, then the constant (constant) is also one. 2) The method of variation of arbitrary constants is used to solve some linear inhomogeneous equations of the second order... Two constants vary here. It is logical to assume that the lesson will consist of two paragraphs…. I wrote this proposal, and for 10 minutes I was painfully thinking what other clever crap to add for a smooth transition to practical examples. But for some reason, there are no thoughts after the holidays, although he did not seem to abuse anything. Therefore, let's go straight to the first paragraph. Variation method of an arbitrary constant Before considering the method of variation of an arbitrary constant, it is advisable to be familiar with the article Linear differential equations of the first order... In that lesson, we practiced first solution non-uniform DE of the 1st order. This first solution, I remind you, is called replacement method or Bernoulli method(not to be confused with Bernoulli equation!!!) We will now consider second solution- method of variation of an arbitrary constant. I will give just three examples, and I will take them from the above lesson. Why so little? Because in fact the solution in the second way will be very similar to the solution in the first way. In addition, according to my observations, the method of variation of arbitrary constants is used less often than the method of replacement. Example 1 Solution: This equation is linear inhomogeneous and has a familiar form: The first step is to solve a simpler equation: In this example, you need to solve the following auxiliary equation: Before us separable equation, the solution of which (hopefully) is no longer difficult for you: Thus: In the second step replace constant of some yet unknown function that depends on "x": Hence the name of the method - we vary the constant. Alternatively, the constant may be some function that we have to find now. V original inhomogeneous equation we will replace: Substitute and into the equation : Control moment - the two terms on the left cancel out... If this does not happen, you should look for the error above. As a result of the replacement, an equation with separable variables is obtained. Separate variables and integrate. What a blessing, exhibitors are also declining: Add the "normal" constant to the found function: At the final stage, we recall our replacement: Function just found! So the general solution is: Answer: common decision: If you print out two solutions, you will easily notice that in both cases we found the same integrals. The only difference is in the solution algorithm. Now for something more complicated, I'll comment on the second example: Example 2 Find the general solution to the differential equation Solution: Let us bring the equation to the form : Let us zero the right-hand side and solve the auxiliary equation:
In the inhomogeneous equation, we make the replacement: According to the rule of product differentiation: Substitute and into the original inhomogeneous equation: The two terms on the left are canceled, which means we are on the right track: We integrate by parts. The tasty letter from the formula for integration by parts has already been used in the solution, so we use, for example, the letters "a" and "be": Now we recall the replacement carried out: Answer: common decision: And one example for a do-it-yourself solution: Example 3 Find a particular solution of the differential equation corresponding to a given initial condition. , The solution at the end of the lesson can serve as a rough example for finishing the assignment. Variation method of arbitrary constants We often heard the opinion that the method of variation of arbitrary constants for a second-order equation is not an easy thing. But I guess the following: most likely, the method seems difficult to many, since it is not so common. But in reality there are no special difficulties - the course of the decision is clear, transparent, understandable. And beautiful. To master the method, it is desirable to be able to solve inhomogeneous second-order equations by selecting a particular solution in the form of the right-hand side. This method is discussed in detail in the article. Inhomogeneous DE of the 2nd order... We recall that the second-order linear inhomogeneous equation with constant coefficients has the form: The selection method, which was considered in the above lesson, works only in a limited number of cases when polynomials, exponents, sines, cosines are on the right side. But what to do when on the right, for example, fraction, logarithm, tangent? In such a situation, the method of variation of constants comes to the rescue. Example 4 Find the general solution of the second order differential equation Solution: On the right side this equation there is a fraction, so we can immediately say that the method of selecting a particular solution does not work. We use the method of variation of arbitrary constants. Nothing foreshadows a thunderstorm, the beginning of the solution is completely ordinary: Find common decision corresponding homogeneous equations: Let's compose and solve the characteristic equation: Pay attention to the record of the general solution - if there are brackets, then we expand them. Now we do practically the same trick as for the first-order equation: we vary the constants, replacing them with unknown functions. That is, general solution to heterogeneous we will seek equations in the form: Where - yet unknown functions. It looks like a junkyard household waste, but now we will sort everything. Derivatives of functions act as unknowns. Our goal is to find derivatives, and the found derivatives must satisfy both the first and second equations of the system. Where do the "games" come from? The stork brings them. We look at the general solution obtained earlier and write down: Find the derivatives: With the left parts sorted out. What's on the right? Is the right side of the original equation, in this case: The coefficient is the coefficient at the second derivative: In practice, almost always, and our example is no exception. Everything cleared up, now you can create a system: The system is usually decided by Cramer's formulas using a standard algorithm. The only difference is that instead of numbers, we have functions. Find main determinant systems: If you have forgotten how the determinant "two by two" is revealed, refer to the lesson How to calculate the determinant? The link leads to the board of shame =) So: this means that the system has a unique solution. Find the derivative: But that's not all, so far we have found only the derivative. Let's deal with the second function:
At the final stage of the solution, we recall in what form we were looking for the general solution of the inhomogeneous equation? In such: The functions you are looking for have just been found! It remains to perform the substitution and write down the answer: Answer: common decision: In principle, the brackets could be expanded in the answer. Full verification of the answer is performed according to the standard scheme, which was discussed in the lesson Inhomogeneous DE of the 2nd order... But the verification will not be easy, since it is necessary to find rather heavy derivatives and carry out a cumbersome substitution. This is an unpleasant feature when you are dealing with diffusion like this. Example 5 Solve a differential equation by varying arbitrary constants This is an example for a do-it-yourself solution. In fact, the right side is also a fraction. We recall the trigonometric formula, by the way, it will need to be applied in the course of the solution. The method of variation of arbitrary constants is the most versatile method. They can solve any equation that is solved by the method of selecting a particular solution by the view of the right side... The question arises, why not use the method of variation of arbitrary constants there too? The answer is obvious: the selection of a private solution, which was considered in the lesson Inhomogeneous second order equations, significantly speeds up the solution and shortens the writing - no fucking with determinants and integrals. Consider two examples with the Cauchy problem. Example 6 Find a particular solution of the differential equation corresponding to the given initial conditions , Solution: Again fraction and exponent in an interesting place. Find common decision corresponding homogeneous equations: General solution to heterogeneous we seek equations in the form:, where - yet unknown functions. Let's compose the system: In this case: Thus: We solve the system using Cramer's formulas: We restore the function by integrating: We restore the second function by integrating: Such an integral is solved variable replacement method: From the replacement itself, we express: Thus: This integral can be found full square selection method, but in the examples with differs, I prefer to expand the fraction method of undefined coefficients: Both functions are found: As a result, the general solution to the inhomogeneous equation is: Let us find a particular solution satisfying the initial conditions . Technically, the search for a solution is carried out in the standard way, which was discussed in the article Inhomogeneous differential equations of the second order. Hold on, now we will find the derivative of the found common solution: Here is such a disgrace. It is not necessary to simplify it; it is easier to immediately compose a system of equations. According to the initial conditions : Substitute the found values of the constants into a general solution: In the answer, the logarithms can be packed a little. Answer: private solution: As you can see, difficulties can arise in integrals and derivatives, but not in the algorithm of the method of variation of arbitrary constants. It was not I who intimidated you, this is all Kuznetsov's collection! For relaxation, a final, simpler example for a do-it-yourself solution: Example 7 Solve the Cauchy problem , An example is simple, but creative, when you make a system, take a close look at it before deciding ;-), Consider now the linear inhomogeneous equation Example # 1. Find the general solution of the equation y "" + 4y "+ 3y = 9e -3 x. Consider the corresponding homogeneous equation y" "+ 4y" + 3y = 0. The roots of its characteristic equation r 2 + 4r + 3 = 0 are equal to -1 and - 3. Therefore, the fundamental system of solutions to the homogeneous equation consists of the functions y 1 = e - x and y 2 = e -3 x. We seek the solution of the inhomogeneous equation in the form y = C 1 (x) e - x + C 2 (x) e -3 x. To find the derivatives C "1, C" 2, we compose the system of equations (8) Example # 2. Solve linear differential equations of the second order with constant coefficients by the method of variation of arbitrary constants: Solution: Since y = C 1 e 4x + C 2 e 2x, then we write the obtained expressions in the form: Let's find a particular solution provided: Substituting x = 0 into the found equation, we get: We get a system of two equations: |
Read: |
---|
New
- Bulychev "Alice's Journey
- Alexey IsaevThe offensive of Marshal Shaposhnikov
- Megaliths of the Empire "Nick Perumov
- Lacks the spirit. "Close in spirit"
- Military police: legal status and powers Officer who performed police functions in the army
- "My Posthumous Adventures" Yulia Voznesenskaya
- Review: Max fry reading order
- Statutes of the Order of the Knights Templar: Charter and Code
- Alexander Sviyash, Yulia Sviyash Smile before it's too late
- Andrey belyanin series chain dogs of the empire