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Problem: The function \( y=\frac{a}{x}+b x \) is given by the table of the approximate values \begin{tabular}{|c|c|c|c|c|} \hline\( x \) & 0.1 & 0.2 & 0.25 & 0.5 \\ \hline\( y \) & \( 29 \cdot N \) & \( 13 \cdot N \) & \( 10 \cdot N \) & \( 2 \cdot N \) \\ \hline \end{tabular} where \( N \) is the number of the variant. Determine the coefficients \( a, b \) by the least squares method. Calculate the value of the root mean square error. Plot the graphs of the function given in the table and the obtained function. \[ N=2 \]

14.3.1 Least square method

3.83 $

Problem: Solve the system of linear algebraic equations by the sweep method: \[ \left\{\begin{array}{c} -4 X_{1}+X_{2}=-5 \\ 2 X_{1}+5 X_{2}-X_{3}=-1 \\ 4 X_{2}-8 X_{3}+3 X_{4}=21 \\ -6 X_{3}-7 X_{4}=-9 \end{array}\right. \]

14.4.1 Sweep method

3.83 $

Problem: Solve the system of equations by the simpleiteration method in 3 steps: \[ \left\{\begin{array}{l} 4 x_{1}-3 x_{2}+x_{3}=7 \\ 2 x_{1}-5 x_{2}-x_{3}=3 \\ -x_{1}+x_{2}-6 x_{3}=5 \end{array}\right. \]

14.5.1 Simple-Iteration method

2.55 $

Problem: Calculate the integral \( \int_{1}^{2} f(x) d x \) with the step \( h=0,25 \), using the formula: a) of central rectangles; estimate the error by the Runge rule and a priori error estimate; б) of trapezoids; estimate the error by the Runge rule and the priori error estimate; clarify the result by Runge; в) Simpson's; estimate the error by the Runge. Intermediate calculations are carried out with six significant figures. Calculate the arguments of trigonometric functions in radians. Take into account the error while writing the answers. \[ \int_{1}^{2} f(x) d x-\text { ?, where } f(x)=e^{x \sqrt{x}} \text {, step } h=0,25 \text {. } \]

14.6.1 Approximate calculation of integrals

6.38 $

Problem: Find a solution to the boundary value problem, using the finite difference method. \( \left\{\begin{array}{c}-y^{\prime \prime}+q(x) y=f(x) \\ y(0)=y_{0}, \quad y(1)=y_{1}\end{array} \quad\right. \) with steps \( h_{1}=\frac{1}{3}, \quad h_{2}=\frac{1}{6} \) and estimate the error by the Runge rule. Plot the graphs of the obtained approximate solutions. \begin{tabular}{|c|c|c|c|c|} \hline \( \mathrm{N} \) & \( q(x) \) & \( f(x) \) & \( y_{0} \) & \( y_{1} \) \\ \hline 11 & 1 & \( 1+6 x-x^{3} \) & 1 & 0 \\ \hline \end{tabular}

14.7.1 Approximate solution of differential equations

6.38 $

Problem: Numerically solve the Cauchy problem on the segment of length 0,8 for a first-order ordinary differential equation \( \left\{\begin{array}{l}y^{\prime}=f(x, y) \\ y(a)=y_{a}\end{array}\right. \), with the step \( h=0,2 \) : a) by the explicit Euler method with an error estimate by the Runge rule; b) by one of Runge-Kutta methods of the \( 2^{\text {nd }} \) order with an error estimate by the Runge rule; c) find the exact solution to the problem.

14.7.2 Approximate solution of differential equations

7.66 $

Problem: Find the solution to the boundary value problem using the finite difference method. \[ \begin{array}{l} y^{\prime \prime}+q(x) y^{\prime}=f(x), \quad x \in[a ; b] \\ y(a)=y_{a}, \quad y(b)=y_{b} \end{array} \] a) with the step \( h=\frac{b-a}{5} \); b) with the step \( h=\frac{b-a}{10} \); c) estimate the error by the Runge rule; d) plot the graphs of the approximate solutions. \[ a=0, b=1 \text {. } \]

14.7.3 Approximate solution of differential equations

8.94 $

Problem: Prove the admissibility of the rule. \[ \frac{\Gamma, A \vdash C, \Gamma, B \vdash C}{\Gamma, A \vee B \vdash C} . \]

6.1.2.1 Sequent calculus

1.53 $

Problem: Construct a derivation of sequence in sequent calculus: \[ (\varphi \rightarrow \psi),(\psi \rightarrow x) \vdash(\varphi \rightarrow x) . \]

6.1.2.2 Sequent calculus

2.55 $

Problem: Prove the derivability of the formula in sequent calculus: \[ \vdash(\overline{A \cdot \bar{B}}) \rightarrow(A \rightarrow B) . \]

6.1.2.3 Sequent calculus

1.53 $

Problem: Prove the derivability of the formula in sequent calculus: \[ \vdash A(\bar{B} \vee C) \rightarrow(A \bar{B} \vee C) \text {. } \]

6.1.2.4 Sequent calculus

1.79 $

Problem: Prove the derivability of the formula in sequent calculus: \[ \vdash(A \rightarrow B) \vee(C \rightarrow B) \rightarrow(A C \rightarrow B) . \]

6.1.2.5 Sequent calculus

1.79 $

Problem: Prove the derivability of the formula in sequent calculus: \[ \vdash \overline{A \vee B} \rightarrow(\mathrm{A} \rightarrow \bar{B}) . \]

6.1.2.6 Sequent calculus

1.53 $

Problem: Find the general solution of the second order inhomogeneous differential equation: \[ 3 y^{\prime \prime}+7 y^{\prime}=-x^{2}+5 \text {. } \]

8.1.2.4 Second order differential equations

1.79 $

Problem: Find the general solution of the second order inhomogeneous differential equation: \[ 2 y^{\prime \prime}-9 y^{\prime}+4 y=\cos 2 x-2 \sin 2 x \text {. } \]

8.1.2.5 Second order differential equations

1.28 $

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