Solution to problem 13.5.2 from the collection of Kepe O.E.

The solution to the differential equation of damped oscillations of a material point is represented by the formula: x = e^(-0.5t) (C1 cos(3t) + C2 sin(3t)). To determine the integration constant C2, it is necessary to use the initial conditions. It is known that at time t0 = 0 the speed of the point is zero, that is, v0 = 0. For this equation, the speed of the point v(t) is equal to the derivative of x(t), that is, v(t) = dx/dt. Substituting x(t) into the expression for speed, we get v(t) = -0.5 e^(-0.5t) (C1 cos(3t) + C2 sin(3t)) + 3 e^(-0.5t ) (C2 cos(3t) - C1 sin(3t)). At t = 0, v(0) = 0, which gives C2 = 0.25. Thus, the integration constant C2 is 0.25.

This digital product is a solution to problem 13.5.2 from the collection of Kepe O.?. The solution to the differential equation of damped oscillations of a material point is presented in the form of a formula that allows you to determine the position of the point depending on time. Along with the solution, initial conditions and a step-by-step explanation of the process of solving the problem are provided. All this is presented in a beautiful html format, which makes it easy to read and allows you to quickly familiarize yourself with the material. This digital product will be useful for anyone who is studying mathematics or needs to solve a similar problem.

The digital product is a solution to problem 13.5.2 from the collection of Kepe O.?. mathematics. This problem is related to the solution of the differential equation of damped oscillations of a material point, which is represented by the formula: x = e^(-0.5t) (C1 cos(3t) + C2 sin(3t)). To determine the integration constant C2, it is necessary to use the initial conditions. It is known that at time t0 = 0 the speed of the point is zero, that is, v0 = 0.

When solving the problem, it is necessary to find the integration constant C2. To do this, use the initial conditions: at t = 0, the speed of the point is zero, that is, v(0) = 0. Substituting x(t) into the expression for speed, we obtain v(t) = -0.5 e^(-0, 5t) (C1 cos(3t) + C2 sin(3t)) + 3 e^(-0.5t) (C2 cos(3t) - C1 sin(3t)). At t = 0, v(0) = 0, which gives C2 = 0.25. Thus, the integration constant C2 is 0.25.

Along with the solution to the problem, initial conditions and a step-by-step explanation of the process of solving the problem are provided. The solution is presented in a beautiful html format, which makes it easy to read and allows you to quickly familiarize yourself with the material. This digital product will be useful for people studying mathematics or needing to solve a similar problem.

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The product in this case is the solution to problem 13.5.2 from the collection of Kepe O.?. mathematics.

This problem consists in solving the differential equation of damped oscillations of a material point, which has the form x = e-0.5t (C1 cos 3t + C2 sin 3t).

It is necessary to determine the integration constant C2, provided that the integration constant C1 is equal to 1.5, and at time t0 = 0 the speed of the point v0 = 0.

To solve this problem, you need to use the initial condition to determine the constants C1 and C2. Thus, the time derivative of the function x is equal to v = -0.5e-0.5t (C1 cos 3t + C2 sin 3t) + 3e-0.5t (-C1 sin 3t + C2 cos 3t).

Substituting t0 = 0 and v0 = 0, we obtain two equations: C1 = 1.5 and -0.5C1 + 3C2 = 0. Solving the system of equations, we find C2 = 0.25.

Thus, the solution to problem 13.5.2 from the collection of Kepe O.?. lies in the found constant of integration C2, equal to 0.25.

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