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Problem: Find all the values of a, for which the system of inequalities \[ \left\{\begin{array}{l} x^{2}+(5 a+2) x+4 a^{2}+2 a<0 \\ x^{2}+a^{2}=4 \end{array}\right. \] has at least one solution.

17.1 USE problems

0 $

Problem: Solve the inequality for all non-negative values of the parameter a: \[ a^{3} x^{4}+2 a^{2} x^{2}-8 x+a+4 \geq 0, \quad a \geq 0 . \]

17.2 USE problems

0 $

Problem: The dynamic intersectoral model of the production system is described by a system of linear homogeneous differential equations: \[ \frac{d x(t)}{d x}=(E-A) B^{-1} \cdot x(t) \text {, где } \frac{d x(t)}{d t}=\left(\frac{d x_{1}}{d t} ; \frac{d x_{2}}{d t} ; \ldots ; \frac{d x_{n}}{d t}\right) \] is the column vector of absolute growth of the production volume; \( x(t)=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \) is the column vector of production volume: \( E \) is the identity matrix: \( A=\left(a_{i j}\right) \) is the matrix of coefficients of direct material costs; \( B=\left(b_{i j}\right) \) is the matrix of coefficients of the capital intensity of the production growth (the costs of production accumulation per unit of production of the corresponding form). Let a two-industry model of a production system be given with a matrix of direct costs \( A=\left(\begin{array}{ll}0,21 & 0,32 \\ 0,32 & 0,45\end{array}\right) \) and the matrix of coefficients of capital intensity of the production growth \( B=\left(\begin{array}{cc}0,52 & 0,52 \\ 1 & 0,83\end{array}\right) \). Find the production volumes of the industries, if at the initial moment \( x(0)=\left(\begin{array}{l}60 \\ 60\end{array}\right) \).

18.1 Mathematical methods and models in economics

5.14 $

Problem: Let's suppose that \( y(t) \) is the national income at time \( t \). The consumption \( c(t) \) lags relative to \( y(t) \) according to the law \( c(t+1)=0,84 y(t)+4 \sin 0,7 t \). An entrepreneur makes investments if the increase of the national income is stable, i.e., \( i(t+2)=1,96(y(t+1)-y(t)) \). An equilibrium state is achieved under the following condition: \( y(t+2)=c(t++2)+i(t+2) \). Find the function of the national income under equilibrium conditions at time \( t \).

18.2 Mathematical methods and models in economics

3.85 $

Problem: The model of intersectoral balance, where the product intended for domestic and final consumption in period \( t \), is determined by the output in the subsequent period \( t+1 \) assuming a constant share of domestic consumption by each industry, will have the form: \[ \left\{\begin{array}{l} x(t+1)=0,5 x(t)+0,1 y(t)+40 \\ y(t+1)=0,4 x(t)+0,5 y(t)+60 \end{array}\right. \] At the initial moment of time the gross output of each of the industries is \( x(0)=100, y(0)=100 \). It is required to calculate the vector of the gross output of each of the industries at the moment \( t=2 \), if the final consumption components increase by \( 30 \% \) for each period.

18.3 Mathematical methods and models in economics

2.57 $

Problem: The dynamics of inflation changes in accordance with the equation \( \frac{d p}{d t}=-\frac{p}{2} \), where \( p(t) \) is the function of the inflation. Determine the trajectory of inflation change.

18.4 Mathematical methods and models in economics

1.28 $

Problem: Let the dynamic function of the supply of the product be equal to the rate of change in the stock of products and given by the equality \( F\left(x_{t}, p_{t}, \frac{d p_{t}}{d t}\right)=\frac{x_{t}}{p_{t}-\sqrt{x_{t} p_{t}}} \frac{d p_{t}}{d t} \), where \( x_{t} \) is the stock of products; \( p_{t} \) is the price of the product \( d s ; \frac{d p_{t}}{d t} \) is the quantity, characterizing the price change. Determine how the price of the product changes depending on its quality.

18.5 Mathematical methods and models in economics

1.28 $

Problem: The state has decided to support the bankrupt enterprise that had stopped the production. During the year, funds will continuously flow into the company's account, moreover the crisis manager has chosen a state support scheme, where the transferred funds gradually decrease from 10 million rubles to 0 by the end of the year. Due to the dilapidation of the equipment, the retirement coefficient of funds per year is 1,5 . The return on investment in this industry is \( 45 \% \). What will the output of the company be in a year?

18.6 Mathematical methods and models in economics

2.57 $

Problem: For the model of inflation dynamics \( p^{\prime \prime}+9 p=0 \) with initial conditions \( p(0)=2, p^{\prime}(0)=6 \). Find the general solution in the following form \( p(t)=R \cos \left(\omega_{0} t-\delta\right) \).

18.7 Mathematical methods and models in economics

1.28 $

Problem: In simple market models, supply and demand are usually supposed to depend only on the price of the product. In real situations, supply and demand also depend on the pricing trend and the rate of price change. In models with continuous and time-differentiable \( t \) functions, these characteristics are described by the first and second derivatives of the cost function \( p(t) \), respectively. Let the function of demand \( D \) and supply \( S \) have the following dependances on the post \( p \) and its derivatives: \( D(t)=p^{\prime \prime}-2 p^{\prime}-6 p+36, S(t)=2 p^{\prime \prime}+ \) \( +2 p^{\prime}+4 p+6 \). Find the dynamics of the equilibrium \( p \) price for the product dependent on time.

18.8 Mathematical methods and models in economics

2.57 $

Problem: A system of differential equations must be composed for the systems, given by the graphs in the figure. \[ \begin{array}{l} P_{0}^{\prime}(t)=-0,01 P_{0}(t) \\ P_{1}^{\prime}(t)=0,01 P_{0}(t)-0,01 P_{2}(t) \\ P_{2}^{\prime}(t)=0,01 P_{1}(t)-0,01 P_{2}(t) \\ P_{3}^{\prime}(t)=0,01 P_{2}(t) \end{array} \text { for the possibilities } P_{i}(t), i=0,1, \ldots \] of the form \( P_{2}^{\prime}(t)=0,01 P_{1}(t)-0,01 P_{2}(t) \) for the possibilities \( P_{i}(t), i=0,1, \ldots \) Wherein, assume that at the initial moment \( (t=0) \) the system is in zero state. a) b) c) d)

15.4.1 Queuing systems

3.85 $

Problem: There are three machines, working in the manufactory, which break down with intensities \( \lambda_{1}, \lambda_{2}, \lambda_{3} \) (per day) respectively. There are two regulators in the staff, eliminating the breakdowns of machines with intensities \( \mu_{1}, \mu_{2} \) (per day) respectively. It is required to plot the graph of this queuing system and find the fraction of time, both regulators are busy with work. \begin{tabular}{|c|c|c|c|c|} \hline\( \lambda_{1} \) & \( \lambda_{2} \) & \( \lambda_{3} \) & \( \mu_{1} \) & \( \mu_{2} \) \\ \hline 0,3 & 0,3 & 0,1 & 0,3 & 0,1 \\ \hline \end{tabular}

15.4.2 Queuing systems

7.71 $

Problem: There are three machines, working in the manufactory, which break down with intensities \( \lambda_{1}, \lambda_{2}, \lambda_{3} \) (per day) respectively. There are two regulators in the staff, eliminating the breakdowns of machines with intensities \( \mu_{1}, \mu_{2} \) (per day) respectively. It is required to plot the graph of this queuing system and find the fraction of time, both regulators are busy with work. \begin{tabular}{|c|c|c|c|c|} \hline\( \lambda_{1} \) & \( \lambda_{2} \) & \( \lambda_{3} \) & \( \mu_{1} \) & \( \mu_{2} \) \\ \hline 0,4 & 0,15 & 0,15 & 0,4 & 0,1 \\ \hline \end{tabular}

15.4.3 Queuing systems

7.71 $

Problem: Find \( \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y} \), if functions \( u \) and \( v \) are implicitly defined by the equalities \[ x=e^{u}+u \sin v, y=e^{u}-u \cos v . \]

2.2.20 Derivatives and differentials

0.77 $

Problem: Let \( u=\frac{1}{x}(\varphi(x-y)+\psi(x+y)) \), where \( \varphi \) and \( \psi \) are differentiable functions. Show that \[ \frac{\partial}{\partial x}\left(x^{2} \frac{\partial u}{\partial x}\right)=x^{2} \frac{\partial^{2} u}{\partial y^{2}} . \]

2.2.21 Derivatives and differentials

1.28 $

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