IDZ 11.3 – Option 7. Solutions Ryabushko A.P.

1. Let's find the general solution to the differential equations: a) y΄΄+ y΄− 6y = 0; Characteristic equation: r^2 + r - 6 = 0 Roots: r1 = -3, r2 = 2 General solution: y(x) = c1e^(-3x) + c2e^(2x)

b) y΄΄+ 9y΄ = 0; Characteristic equation: r^2 + 9 = 0 Roots: r1 = -3i, r2 = 3i General solution: y(x) = c1cos(3x) + c2sin(3x)

c) y΄΄− 4y΄+ 20y = 0 Characteristic equation: r^2 - 4r + 20 = 0 Roots: r1 = 2i, r2 = -2i General solution: y(x) = c1e^(2ix) + c2e^(-2ix) = c1cos(2x) + c2sin(2x) + i(c1sin(2x) - c2cos(2x))

1. Let's find the general solution of the differential equation: y΄΄+ y = 2cos(x) - (4x + 4)sin(x) Characteristic equation: r^2 + 1 = 0 Roots: r1 = i, r2 = -i General solution of the homogeneous equation : y(x) = c1cos(x) + c2sin(x) Particular solution of the inhomogeneous equation: y*(x) = -2x*cos(x) - 2sin(x)

2. Let's find the general solution of the differential equation: y΄΄+ 2y΄+ y = 4x^3 + 24x^2 + 22x - 4 Characteristic equation: r^2 + 2r + 1 = 0 Root of multiplicity 2: r = -1 General solution of the homogeneous equation : y(x) = (c1 + c2*x)e^(-x) Particular solution of the inhomogeneous equation: y(x) = x^3 + 6x^2 + 5x - 1

3. Let us find a particular solution to the differential equation that satisfies the given initial conditions: y΄΄- 4y΄ + 20y = 16xe^(2x), y(0) = 1, y΄(0) = 2 Characteristic equation: r^2 - 4r + 20 = 0 Roots: r1 = 2 + 4i, r2 = 2 - 4i General solution of the homogeneous equation: y(x) = c1*e^(2x)cos(4x) + c2e^(2x)sin(4x) Particular solution of the inhomogeneous equation: y(x) = (1/4)xe^(2x) - (1/8)*e^(2x) + (3/8)*cos(4x) + (5/32)*sin(4x)

4. Let us define and write the structure of a particular solution y* of a linear inhomogeneous differential equation according to the form of the function f(x): y΄΄- 3y΄ + ​​2y = f(x); a) f(x) = x + 2e^x; Let's find the general solution of the homogeneous equation: r^2 - 3r + 2 = 0 Roots: r1 = 1, r2 = 2 General solution of the homogeneous equation: y(x) = c1e^x + c2e^(2x) A particular solution to an inhomogeneous equation can be sought by the method of indefinite coefficients. Suppose that y*(x) has the form: y*(x) = Ax + Be^x Then y΄(x) = A + Be^x, y΄΄(x) = Be^x Substitute into the original equation and find the values ​​of the coefficients: A = -2, B = 1 Particular solution of the inhomogeneous equation: y(x) = -2x + e^x

b) f(x) = 3cos(4x) Find the general solution of the homogeneous equation: r^2 - 3r + 2 = 0 Roots: r1 = 1, r2 = 2 General solution of the homogeneous equation: y(x) = c1e^x + c2e^(2x) A particular solution to an inhomogeneous equation can be sought by the method of varying constants. Let us assume that the particular solution has the form y*(x) = Acos(4x) + Bsin(4x). Then y΄(x) = -4Asin(4x) + 4Bcos(4x), y΄΄(x) = -16Acos(4x) - 16Bsin(4x). We substitute into the original equation and find the values ​​of the coefficients: A = 0, B = -3/17 Particular solution of the inhomogeneous equation: y*(x) = (-3/17)*sin(4x)

IDZ 11.3 – Option 7. Solutions Ryabushko A.P. is a digital product that represents solutions to problems in mathematics (option 7) for completing individual homework. In this product you will find a complete and detailed solution to each problem, made by an experienced teacher A.P. Ryabushko. Each solution is accompanied by detailed calculations, explanations and graphic illustrations, which makes this product ideal for self-preparation for an exam or test in mathematics.

The HTML design of the product is made in a beautiful and clear style, which provides a convenient and intuitive interface for users. You can easily find the problem you need and study its solution using convenient links and page navigation. Thanks to this, the product becomes an indispensable assistant for students and schoolchildren who strive to improve their knowledge in mathematics.

IDZ 11.3 – Option 7. Solutions Ryabushko A.P. is a digital product consisting of solutions to problems in mathematics, including solutions to the following tasks:

1. Find the general solution to the differential equation: a) y΄΄+ y΄− 6y = 0; b) y΄΄+ 9y΄ = 0; c) y΄΄− 4y΄+ 20y = 0

2. Find the general solution to the differential equation: y΄΄+ y = 2cosx – (4x + 4)sinx

3. Find the general solution to the differential equation: y΄΄+ 2y΄+ y = 4x3 + 24x2 + 22x – 4

4. Find a particular solution to the differential equation that satisfies the given initial conditions: y΄΄− 4y΄ + 20y = 16xe2x, y(0) = 1, y΄(0) = 2

5. Determine and write the structure of a particular solution y* of a linear inhomogeneous differential equation based on the form of the function f(x) 5.7 y΄΄− 3y΄ + ​​2y = f(x); a) f(x) = x + 2ex; b) f(x) = 3cos4x

Each solution contains detailed calculations, explanations and graphic illustrations made by an experienced teacher A.P. Ryabushko. The HTML design of the product is made in a beautiful and clear style, providing a convenient and intuitive interface for users. This product may be useful for students and schoolchildren who want to improve their knowledge in mathematics and prepare for exams or tests.

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IDZ 11.3 – Option 7. Solutions Ryabushko A.P. is a set of solutions to differential equations consisting of five problems.

The first problem requires finding a general solution to a differential equation of the form y΄΄+ y΄− 6y = 0, the second problem - the form y΄΄+ 9y΄ = 0, and the third problem - the form y΄΄− 4y΄+ 20y = 0.

The fourth problem requires finding a particular solution to the differential equation y΄΄− 4y΄ + 20y = 16xe2x, which satisfies the initial conditions y(0) = 1 and y΄(0) = 2.

The fifth problem requires determining and writing down the structure of a particular solution y* of the linear inhomogeneous differential equation y΄΄− 3y΄ + ​​2y = f(x), where the function f(x) is given as a) f(x) = x + 2ex and b ) f(x) = 3cos4x.

All solutions to problems are prepared in Microsoft Word 2003 using a formula editor and contain detailed mathematical calculations.

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