IDZ 11.3 – Option 13. Solutions Ryabushko A.P.

Let's find the general solution to the differential equations: a) 9y΄΄+ 6y΄ + y = 0; Characteristic equation: 9λ² + 6λ + 1 = 0 D = 36 - 4*9 = 0 λ₁ = λ₂ = -1/3 y₁(x) = e^(-x/3) y₂(x) = xe^(-x /3) Y(x) = c₁e^(-x/3) + c₂xe^(-x/3)

b) y΄΄− 4y΄− 21y = 0; Characteristic equation: λ² - 4λ - 21 = 0 D = 16 + 84 = 100 λ₁ = 7, λ₂ = -3 y₁(x) = e^(7x) y₂(x) = e^(-3x) Y(x) = c₁e^(7x) + c₂e^(-3x)

c) y΄΄+ y = 0 Characteristic equation: λ² + 1 = 0 D = -4 λ₁ = i, λ₂ = -i y₁(x) = cos(x) y₂(x) = sin(x) Y(x ) = c₁cos(x) + c₂sin(x)

Let's find the general solution of the differential equation: y΄΄ – 8y΄ + 12y = 36x^4 – 96x^3 + 24x^2 + 16x – 2 Characteristic equation: λ² - 8λ + 12 = 0 D = 64 - 48 = 16 λ₁ = 6 , λ₂ = 2 y₁(x) = e^(6x) y₂(x) = e^(2x) Yh(x) = c₁e^(6x) + c₂e^(2x)

Let us find a particular solution y_p(x) of the inhomogeneous equation by the method of indefinite coefficients: y_p(x) = Ax^4 + Bx^3 + Cx^2 + Dx + E y΄(x) = 4Ax^3 + 3Bx^2 + 2Cx + D y΄΄(x) = 12Ax^2 + 6Bx + 2C Substitute into the equation: 12Ax^2 + 6Bx + 2C - 8(4Ax^3 + 3Bx^2 + 2Cx + D) + 12(Ax^4 + Bx^3 + Cx^2 + Dx + E) = 36x^4 – 96x^3 + 24x^2 + 16x – 2 12A = 36, -32A + 6B = -96, 20A - 24B + 2C = 24, -8A + 12B - 8C + 2D = 16, 12A - 8B + 12C - 8D + 12E = -2 A = 3, B = 9/4, C = -27/8, D = -21/16, E = -131/288 Yp( x) = 3x^4 + (9/4)x^3 - (27/8)x^2 - (21/16)x - 131/288 Y(x) = Yh(x) + Yp(x)

Let's find the general solution of the differential equation: y΄΄ + 4y΄ + 4y = 6e^(-2x) Characteristic equation: λ² + 4λ + 4 = 0 D = 16 - 16 = 0 λ₁ = λ₂ = -2 y₁(x) = e ^(-2x) y₂(x) = xe^(-2x) Yh(x) = c₁e^(-2x) + c₂xe^(-2x)

Let us find a particular solution y_p(x) of the inhomogeneous equation by the method of variation of constants: Let us present the solution in the form y(x) = u(x)e^(-2x) y΄(x) = u΄(x)e^(-2x) - 2u(x)e^(-2x) y΄΄(x) = u΄΄(x)e^(-2x) - 4u΄(x)e^(-2x) + 4u(x)e^(- 2x) Substitute into the equation: u΄΄(x)e^(-2x) + 2u΄(x)e^(-2x) = 6 Integrate both sides: u΄(x)e^(-2x) = 3x + c₁ u(x) = -3/2x^2 - c₁/2x + c₂ Yp(x) = (-3/2x^2 - c₁/2x + c₂)e^(-2x) Y(x) = Yh( x) + Yp(x)

Let us find a particular solution to the differential equation that satisfies the given initial conditions: y΄΄− 6y΄ + 25y = (32x – 12)sin(3x) – 36xcos(3x), y(0) = 4, y΄(0) = 0 Characteristic equation: λ² - 6λ + 25 = 0 D = 36 - 100 = -64 λ₁ = 3 - 4i, λ₂ = 3 + 4i y₁(x) = e^(3x)sin(4x) y₂(x) = e^( 3x)cos(4x) Yh(x) = e^(3x)(c₁sin(4x) + c₂cos(4x))

Let us find a particular solution y_p(x) of the inhomogeneous equation by the method of indefinite coefficients: y_p(x) = A(x)sin(3x) + B(x)cos(3x) y΄(x) = 3A(x)cos(3x) - 3B(x)sin(3x) + 32 y΄΄(x) = -9A(x)sin(3x) - 9B(x)cos(3x) Substitute into the equation: -9A(x)sin(3x) - 9B (x)cos(3x) - 6(3A(x)cos(3x) - 3B(x)sin(3x) + 32) + 25(A(x)sin(3x) + B(x)cos(3x) ) = (32x - 12)sin(3x) - 36xcos(3x) -18A(x) + 25A(x) = -36x, -18B(x) + 25B(x) = 32 A(x) = 4x/7 , B(x) = 32/7 Yp(x) = (4x/7)sin(3x) + (32/7)cos(3x) Y(x) = Yh(x) + Yp(x) Substitute the initial conditions and find the constants: Y(0) = c₂ = 4 Y΄(0) = 3c₁ + 32/7 = 0 => c₁ = -32/21 Y(x) = e^(3x)(-4/3sin(4x ) + 32/21cos(4x)) + (4x/7)sin(3x) + (32/7)cos(3x)

Let us define and write the structure of a particular solution y*(x) of a linear inhomogeneous differential equation by the form of the function f(x): y΄΄– 6y΄ + 9y = f(x) a) f(x) = (x – 2)e^ (3x) Characteristic equation: λ² - 6λ + 9 = 0 (λ - 3)² = 0 => λ₁ = λ₂ = 3 y₁(x) =

This product is a digital product offered in the Digital Product Store. This is a solution to problem IDZ 11.3, Option 13, prepared by the author Ryabushko A.P.

Solution IDZ 11.3 - Option 13 includes a detailed description of the solution to the problem, as well as a step-by-step explanation of all the steps required to solve it.

The product is designed in a beautiful html format, which makes it easy to read and understand. All formulas and graphs are presented in a clear and visual form, which makes it easy to understand the solution to the problem and follow each step of the solution.

This product is an excellent choice for those who are looking for high-quality and detailed material for preparing for exams or doing homework in the field of mathematics.

This product is a solution to the problem IDZ 11.3 – Option 13, which includes a detailed description of the solution to differential equations and methods for solving them. The solution was prepared by the author Ryabushko A.P. and is designed in a beautiful html format, which makes it easy to read and understand.

The product includes a step-by-step explanation of all the steps required to solve the problem, and also provides all the necessary formulas and graphs in a clear and visual manner.

This product may be useful to students and teachers studying differential equations and their solutions. It is an excellent choice for those looking for high-quality and detailed material for studying this topic.

***

IDZ 11.3 – Option 13. Solutions Ryabushko A.P. is a set of solutions to differential equations consisting of five problems. Each problem is a differential equation that needs to be solved.

In the first problem, you need to find a general solution to the differential equation, which has the form 9y΄΄+ 6y΄ + y = 0 in point a), the form y΄΄− 4y΄− 21y = 0 in point b) and the form y΄΄+ y = 0 in point c).

In the second problem, it is necessary to find a general solution to the differential equation, which has the form y΄΄ – 8y΄ + 12y = 36x4 – 96x3 + 24x2 + 16x – 2.

The third problem requires finding a general solution to the differential equation, which has the form y΄΄ + 4y΄ + 4y = 6e–2x.

In the fourth problem, it is necessary to find a particular solution to the differential equation, which has the form y΄΄− 6y΄ + 25y = (32x – 12)sin3x – 36xcos3x, provided that y(0) = 4 and y΄(0) = 0.

In the fifth problem, it is required to determine and write down the structure of a particular solution y* of a linear inhomogeneous differential equation in terms of the form of the function f(x), which has the form y΄΄– 6y΄ + 9y = f(x). In point a) the function f(x) is equal to (x – 2)e3x, and in point b) the function f(x) is equal to 4cosx.

Each problem has a detailed solution, designed in Microsoft Word 2003 using the formula editor.

***

Amazing digital product! Solutions to problems in option 13 of IDZ 11.3 are highly accurate and clear.
I am very pleased with the purchase of this digital product. Decisions Ryabushko A.P. helped me successfully complete the tasks of option 13 of IDZ 11.3.
Cool digital product! Solutions to problems in version 13 of IDZ 11.3 are easy to use and allow you to quickly and efficiently complete your homework.
This digital item is worth every penny! Decisions Ryabushko A.P. helped me deeply understand the material of option 13 of IDZ 11.3.
Great digital product! Solutions to problems of IDZ 11.3 – Option 13 from the author Ryabushko A.P. are an indispensable assistant for all students.
I would happily recommend this digital product to all my friends and acquaintances. Decisions Ryabushko A.P. is the best way to successfully complete your math homework.
This digital product exceeded my expectations! Solutions to problems of option 13 IDZ 11.3 from Ryabushko A.P. helped me better understand the material and prepare for the exam.