IDZ 11.3 – Option 22. Solutions Ryabushko A.P.

1. Find the general solution of the differential equation: 1.22 a) y'' - 6y' + 13y = 0; b) y'' - 2y' - 15y = 0; c) y'' - 8y' = 0.

   Solution: a) The characteristic equation λ^2 - 6λ + 13 = 0 has complex roots λ1 = 3 - 2i and λ2 = 3 + 2i. The general solution has the form y = e^3x(C1cos2x + C2sin2x), where C1, C2 are arbitrary constants. b) The characteristic equation λ^2 - 2λ - 15 = 0 has roots λ1 = -3 and λ2 = 5. The general solution has the form y = C1e^-3x + C2e^5x, where C1, C2 are arbitrary constants. c) The characteristic equation λ^2 - 8λ = 0 has roots λ1 = 0 and λ2 = 8. The general solution has the form y = C1 + C2e^8x, ​​where C1, C2 are arbitrary constants.

2. Find the general solution to the differential equation: 2.22 y'' - 4y' + 5y = (24sinx + 8cosx)e^-2x.

   Solution: The characteristic equation λ^2 - 4λ + 5 = 0 has roots λ1 = 2 - i and λ2 = 2 + i. A particular solution can be sought in the form y* = Ae^-2x(sin x + bcos x), where A and b are unknown coefficients. Substituting this form into the original equation, we find A = 4 and b = -2. Then the general solution has the form y = C1e^(2- i)x + C2e^(2 + i)x + 4e^-2x(sin x - 2cos x), where C1, C2 are arbitrary constants.

3. Find the general solution of the differential equation: 3.22 y'' + 4y' = 15e^x.

   Solution: The characteristic equation λ^2 + 4λ = 0 has roots λ1 = -2i and λ2 = 2i. A particular solution can be sought in the form y* = Ax^2e^x, where A is the unknown coefficient. Substituting this form into the original equation, we find A = 15/2. Then the general solution has the form y = C1cos(2x) + C2sin(2x) + (15/2)x^2e^x, where C1, C2 are arbitrary constants.

4. Find a particular solution to the differential equation that satisfies the given initial conditions: 4.22 y'' + 12y' + 36y = 72x^3 - 18, y(0) = 1, y'(0) = 0.

   Solution: The characteristic equation λ^2 + 12λ + 36 = 0 has a root λ = -6, which has a multiplicity of 2. Then the general solution is y = (C1 + C2x)e^(-6x). A particular solution can be sought in the form y*= Ax^3 + Bx^2 + Cx + D, where A, B, C, D are unknown coefficients. Substituting this form into the original equation, we find A = 2, B = -3, C = -1 and D = 1. Then the partial solution has the form y* = 2x^3 - 3x^2 - x + 1. The final solution has the form y = (C1 + C2x)e^(-6x) + 2x^3 - 3x^2 - x + 1, where C1, C2 are arbitrary constants.

5. Determine and write down the structure of a particular solution y* of a linear inhomogeneous differential equation based on the form of the function f(x): 5.22 y'' - 2y' - 15y = f(x); a) f(x) = 4xe^(3x); b) f(x) = x*sin(5x).

   Solution: a) A particular solution can be sought in the form y* = x(Axe^(3x) + B), where A, B are unknown coefficients. Substituting this form into the original equation, we find A = 1/18 and B = -1/75. Then the structure of the particular solution has the form y* = x((1/18)xe^(3x) - 1/75). b) A particular solution can be sought in the form y* = Axsinx + Bxcos5x, where A, B are unknown coefficients. Substituting this form into the original equation, we find A = -1/2 and B = 0. Then the structure of the particular solution has the form y* = (-1/2)xsinx.

IDZ 11.3 – Option 22. Solutions Ryabushko A.P. is a digital product presented in a digital goods store. This product includes solutions to problems in mathematical analysis, compiled by the author Ryabushko A.P. for option 22 of task 11.3.

The design of this digital product is made in a beautiful html format, which makes using the solutions more convenient and enjoyable. The product also includes detailed and clear explanations that will help you understand the principles of problem solving and formulate your own solutions.

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IDZ 11.3 – Option 22. Solutions Ryabushko A.P. is a digital product presented in a digital goods store. This product contains solutions to five problems in mathematical analysis, compiled by the author Ryabushko A.P. for option 22 of task 11.3.

Solutions to problems are presented in the form of detailed explanations that will help you understand the principles of solving differential equations. Each solution is designed in a beautiful html format, which makes using the solutions more convenient and enjoyable.

The first three problems (1.22 a), b), c)) consist of finding a general solution to the differential equation. In each case, the characteristic equation and its roots are given, on the basis of which the general solution is found.

The fourth problem (4.22) is to find a particular solution to the differential equation that satisfies the given initial conditions. The general solution of the differential equation is given, then a particular solution is found using the method of indefinite coefficients.

The fifth task (5.22 a), b)) is to determine and write the structure of a particular solution y* of a linear inhomogeneous differential equation for a given form of function f(x). In each case, the type of particular solution and the coefficients are given, which are found using the method of indefinite coefficients.

IDZ 11.3 – Option 22. Solutions Ryabushko A.P. is an excellent choice for students and teachers who are involved in mathematical analysis and want to improve their knowledge and skills in this area. Such a digital product is an indispensable assistant in preparing for exams and passing assignments.

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IDZ 11.3 – Option 22. Solutions Ryabushko A.P. is a collection of solutions to differential equations, consisting of five problems. Each problem requires finding a general or particular solution to a differential equation with given initial conditions or determining the structure of a particular solution to a linear inhomogeneous differential equation based on the form of the function f(x).

The first task consists of three points in which it is required to find a general solution to differential equations of various types. The second problem also requires finding a general solution to a differential equation, but this time inhomogeneous, with the right-hand side given as a sum of trigonometric functions multiplied by an exponent.

The third problem requires finding a general solution to a differential equation with the right-hand side given as an exponential. The fourth task is to find a particular solution to a linear homogeneous differential equation with the right-hand side in the form of a third-degree polynomial and given initial conditions.

The fifth problem also requires finding a particular solution to a linear inhomogeneous differential equation with the right-hand side specified as a function f(x), but in two versions: with a polynomial and with the product of a sine and a variable. Solutions to all problems are presented in Microsoft Word 2003 format using the formula editor.

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