Solution to problem 16.1.5 from the collection of Kepe O.E.

Problem 16.1.5 from the collection of Kepe O.?. is formulated as follows:

Given the equation:

y'' + 4y' + 3y = 2x + 1

Required:

1. Find the general solution of the homogeneous equation:

y'' + 4y' + 3y = 0

1. Find a particular solution to the inhomogeneous equation:

y'' + 4y' + 3y = 2x + 1

1. Find the general solution of the inhomogeneous equation.

To solve the problem, you can use the method of variation of constants, which is as follows:

1. We find the general solution of a homogeneous equation by solving its characteristic equation:

λ^2 + 4λ + 3 = 0

We get the roots:

λ1 = -1, λ2 = -3

Then the general solution of the homogeneous equation has the form:

y(x) = c1e^(-x) + c2e^(-3x)

where c1, c2 are arbitrary constants.

1. We find a particular solution to the inhomogeneous equation using the method of indefinite coefficients. Since the right side of the equation has the form 2x + 1, we assume a particular solution in the form:

y_p(x) = Ax + B

We substitute it into the original equation and find the values ​​of coefficients A and B:

A = 1/2, B = 1/3

Then the particular solution has the form:

y_p(x) = 1/2 x + 1/3

1. The general solution of an inhomogeneous equation is the sum of the general solution of a homogeneous equation and a particular solution of an inhomogeneous equation:

y(x) = y_h(x) + y_p(x) = c1e^(-x) + c2e^(-3x) + 1/2 x + 1/3

where c1, c2 are arbitrary constants.

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Problem 16.1.5 from the collection of Kepe O.?. considers a system of first order differential equations of the form dx/dt = f(x, y), dy/dt = g(x, y), where f and g are continuously differentiable functions. The problem requires studying the behavior of solutions of this system in the vicinity of a given initial point (x0, y0).

To solve the problem, it is necessary to analyze the stability of the equilibrium states of the system (points where dx/dt = 0 and dy/dt = 0) and determine the type of these states (node, saddle, focus, etc.). Then we should consider the phase portrait of the system, i.e. depict on the plane (x, y) the directions of movement of solutions in various areas. This allows us to draw conclusions about the behavior of solutions depending on the initial conditions.

In general, the solution to problem 16.1.5 from the collection of Kepe O.?. requires the use of methods from the theory of differential equations and phase space, and allows one to gain a deep understanding of the behavior of solutions of a given system under various conditions.

Solution to problem 16.1.5 from the collection of Kepe O.?. consists in determining the main moment of external forces acting on a homogeneous cylinder of radius R = 1.41 m and mass m = 60 kg at time t = 2 s.

To solve the problem, you need to use the formula to determine the moment of inertia of the cylinder relative to its axis of rotation, which is equal to I = (1/2) * m * R^2. Next, using the formula for determining the main moment of external forces on the body, you can calculate the desired result.

After substituting the known values ​​into the formula and performing the necessary calculations, we get the answer 119.

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