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Problem: Bring the equation to canonical form and find its general solution: \[ a_{11} u_{x x}^{\prime \prime}+2 a_{12} u_{x y}^{\prime \prime}+a_{22} u_{y y}^{\prime \prime}+a_{10} u_{x}^{\prime}+a_{20} u_{y}^{\prime}=0 \] \begin{tabular}{|c|c|c|c|c|} \hline\( a_{11} \) & \( a_{12} \) & \( a_{22} \) & \( a_{10} \) & \( a_{20} \) \\ \hline 1 & 4 & 12 & \( 1 / 6 \) & 1 \\ \hline \end{tabular}

11.5.4.1 With constant coefficients

6.36 $

Problem: Bring the second order equation to canonical form and determine its type: \[ 2 u_{x x}+3 u_{x y}+u_{y y}+7 u_{x}+4 u_{y}-2 u=0 . \]

11.5.4.2 With constant coefficients

4.32 $

Problem: Solve the mixed problem: \[ \begin{array}{l} u_{t t}=u_{x x}+5 u+t, 00, \\ \left.u\right|_{x=0}=0,\left.u\right|_{x=\pi}=0, \\ \left.u\right|_{t=0}=\sin 4 x,\left.u_{t}\right|_{t=0}=0 . \end{array} \]

11.5.5.1 Mixed problems

7.63 $

Problem: Solve the mixed problem: \[ \left\{\begin{array}{c} u_{t t}=u_{x x}+t^{2} x, \quad t>0 \\ \left.u\right|_{t=0}=x^{2},\left.\quad u_{t}\right|_{t=0}=0 \end{array}\right. \]

11.5.5.2 Mixed problems

3.05 $

Problem: Solve the mixed problem: \[ \left\{\begin{array}{c} u_{t t}=u_{x x}+6 t, \quad t>0, \quad x>0 \\ \left.u\right|_{t=0}=2 x,\left.\quad u_{t}\right|_{t=0}=0,\left.\quad u\right|_{\chi=0}=t^{3} \end{array}\right. \]

11.5.5.3 Mixed problems

3.81 $

Problem: For the given functions \( \varphi \) and \( \psi \) : a) plot the graph of the functions \( \varphi \) and \( \psi \); b) calculate the convolution \( \varphi * \psi \) of the functions \( \varphi \) and \( \psi \); c) plot the graph of the convolution \( \varphi * \psi \). The function \( \varphi \) is given by the formula, the graph of the function \( \psi \) is broken-line, connecting the points \( A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right), D\left(x_{4}, y_{4}\right) \) (outside the segment \( \left[x_{1}, x_{4}\right] \) the function is equal to zero). \[ \varphi(x)=\operatorname{rect} x=\eta\left(\frac{1}{2}-|x|\right)=\left\{\begin{array}{c} 1,-\frac{1}{2}\frac{1}{2} \end{array}\right. \] where \( \eta(t)=\left\{\begin{array}{l}0, t<0 \\ 1, t \geq 0\end{array}\right. \) is Heaviside step function \[ \begin{array}{l} A(-2,0), B(-1,2), C(1,2), D(2,0), \quad \psi: A B C D \text { (broken - line), } \\ \psi(x)=0 \text { when } x \notin[-2,2] . \end{array} \]

11.3.1 Convolution of functions

10.17 $

Problem: Find the convolution of the functions \( f(x) \) and \( g(x) \), if the function \( f(x) \) takes a value, equal to zero, when \( x \notin[-1,4] \), and when \( x \in[-1,4] \) its graph consists of links of the broken-line \( A B C D \) : \[ \begin{array}{l} A(-1,0), B(1,2), C(2,-2), D(4,-2), \\ g(x)=\left\{\begin{array}{cc} 0, & x<0 \\ 1, & 0 \leq x<1 \\ 0, & x \geq 1 \end{array}\right. \end{array} \]

11.3.2 Convolution of functions

7.63 $

Problem: For piecewise constant functions \( f(x) \) and \( g(x) \) of the form \[ \begin{array}{c} f(x)=\left\{\begin{array}{cc} 0, & x<-1 \\ 1, & -1 \leq x<0 \\ -2, & 0 \leq x<2 \\ 0, & x \geq 2 \end{array}\right. \\ g(x)=\left\{\begin{array}{cc} 0, & x<0 \\ 1, & 0 \leq x<1 \\ -2, & 1 \leq x<3 \\ 0, & x \geq 3 \end{array}\right. \end{array} \] find the cross-covariance and cross-correlation functions.

11.3.3 Convolution of functions

8.9 $

Problem: On the sides of the triangle \( A B C \) points \( A_{1} \in \) \( [B C], A_{2} \in\left[A_{1} C\right], B_{1} \in[C A], B_{2} \in\left[B_{1} A\right] \) are marked. \( C_{1} \in[A B], C_{2} \in\left[C_{1} B\right] \), for which \[ \begin{array}{l} \frac{C A_{1}}{B C}=\frac{C B_{2}}{C A}=\frac{B C+C A}{A B+B C+C A}, \\ \frac{A B_{1}}{C A}=\frac{A C_{2}}{A B}=\frac{C A+A B}{A B+B C+C A}, \\ \frac{B C_{1}}{A B}=\frac{B A_{2}}{B C}=\frac{A B+B C}{A B+B C+C A} . \end{array} \] Prove that the intersection points of the lines \( A_{1} C_{2}, C_{1} B_{2} \) and \( B_{1} A_{2} \) line on the circumscribed circle of the triangle \( A B C \).

12.1.1 Olympic geometry

10.17 $

Problem: In a trapezoid the ratio of the bases is \( 1: 3 \), and the diagonals 2:3. The lines, drawn through sides, are perpendicular. Find the ratio of the lengths of the sides.

12.1.2 Olympic geometry

3.05 $

Problem: Find a six-digit number, the products of which by \( n(n=2,3,4,5,6) \) give numbers in arbitrary order, obtained from the desired by the principle of circular replacement.

12.2.1 Number theory

3.81 $

Problem: Find all 4-digit numbers \( a b c d \) (where \( a, b, c, d \) are decimal digits), each of which is a divisor of at least one of the three four-digit numbers \( b c d a, c d a b \), dabc composed of it.

12.2.2 Number theory

7.63 $

Problem: Let \( m, n \) be numbers greater than 1 . Prove that \( m^{n} \) is presented as a sum of consecutive odd numbers.

12.2.3 Number theory

3.05 $

Problem: Round the number \( a=0,5784 \) to correct digits and calculate the absolute error of the result.

14.1.1 Approximate numbers

1.27 $

Problem: Find the minimum of the function \( y=x^{2}+4 x \) on the segment \( [-3 ; 0] \) using the golden-section search method. Make three iterations.

14.2.1 Golden section search method

2.54 $

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