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Problem: Determine the type of the \( 2^{\text {nd }} \) order equation and bring it to canonical form: \[ u_{x x}+10 u_{x y}+25 u_{y y}=0 . \]

11.5.4.7 With constant coefficients

1.78 $

Problem: Determine the type of the \( 2^{\text {nd }} \) order equation and bring it to canonical form. \[ u_{x x}-2 u_{x y}-8 u_{y y}+u_{x}-u_{y}=0 . \]

11.5.4.8 With constant coefficients

1.78 $

Problem: Determine the type of the second order equation and bring it to canonical form: \[ u_{x x}-9 u_{y y}+3 u_{y}=0 . \]

11.5.4.9 With constant coefficients

1.78 $

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

11.5.4.10 With constant coefficients

1.78 $

Problem: Bring the equation to canonical form, determine its type and find its particular solution, satisfying the given conditions: \[ u_{x y}+2 u_{x}=0,\left.\quad u\right|_{y=-x}=1,\left.\quad u_{y}^{\prime}\right|_{y=-x}=x . \]

11.5.4.11 With constant coefficients

2.54 $

Problem: Determine the type of the equation and find the general solution: \[ 2 u_{x x}+u_{x y}-u_{y y}=0 . \]

11.5.4.12 With constant coefficients

2.54 $

Problem: Let \( \left(X, A, \mu\right. \) ) be a measurable space, \( A_{n} \in A, n \in \mathbb{N} \) is the sequence of such measurable sets that \[ \sum_{n=1}^{\infty} \mu\left(A_{n}\right)<\infty . \] Prove that \( \lim \sup _{n \rightarrow \infty} A_{n} \) is measurable and its measure is equal to 0 .

19.6.3.1 Convergence (in measure, almost everywhere)

3.81 $

Problem: Bring an example of a sequence of functions, converging in measure but not converging almost everywhere.

19.6.3.2 Convergence (in measure, almost everywhere)

5.09 $

Problem: Bring an example of a sequence of such functions \( f_{n} \), satisfying the conditions of Fatou's lemma, that \[ \lim _{n \rightarrow \infty} \int_{A} f_{n}(x) \mu(d x): \] a) doesn't exist, b) exists and isn't equal to \( \int_{A} \lim _{n \rightarrow \infty} f_{n}(x) \mu(d x) \).

19.6.3.3 Convergence (in measure, almost everywhere)

5.09 $

Problem: Find the solution of the second order linar equation, satisfying the given initial conditions (the Cauchy problem). \[ \begin{array}{l} y^{2} z_{x x}+2 y \cdot z_{x y}+z_{y y}+z_{x}=0, \\ z(x ; 0)=x^{3}, \quad z_{y}(x ; 0)=-x . \end{array} \]

11.5.3.5 With variable coefficients

6.36 $

Problem: Solve the mixed problem, using the Fourier method: \[ \left\{\begin{array}{l} u_{t}=u_{x x}, 00 \\ u(x, 0)=\cos ^{4} x-\sin ^{4} x \\ u(0, t)=u_{x}(\pi, t)=0 \end{array}\right. \]

11.5.5.4 Mixed problems

7.63 $

Problem: Solve the problem with stationary inhomogeneities using the Fourier method: \[ \begin{array}{ll} u_{t t}=u_{x x}+6 x, & u(x ; 0)=\sin ^{2} x-x^{3}, \\ u_{t}=(x ; 0)=0, & u(0 ; t)=0, \quad u(\pi ; t)=-\pi^{3} . \end{array} \]

11.5.5.5 Mixed problems

6.36 $

Problem: Solve the wave equation on the given segment, with the boundary conditions \( V(0, t)==V(l, t)=0 \) and given initial conditions, using the Fourier method: \[ \begin{array}{l} V_{t t}^{\prime \prime}=121 V_{x x}^{\prime \prime} \\ V(x, 0)=\left\{\begin{array}{l} \frac{x}{7}, \quad 0 \leq x \leq 14 \\ \frac{28-x}{7}, \quad 14 \leq x \leq 28 \end{array} \quad V_{t}^{\prime}(x, 0)=0 .\right. \end{array} \]

11.5.2.4 Fourier method

7.63 $

Problem: Solve the equation \( u_{x x}-\frac{1}{a^{2}} u_{t t}=0,(0 \leq x \leq l) \) under the given initial and boundary conditions. \begin{tabular}{|c|c|c|c|} \hline\( u(0, t) \) & \( u(l, t) \) & \( u(x, 0) \) & \( u_{t}^{\prime}(x, 0) \) \\ \hline 0 & 0 & \( 3 \sin \frac{5 \pi}{l} x \) & \( 7 x \) \\ \hline \end{tabular}

11.5.2.5 Fourier method

7.63 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=1, \quad u(0, x)=\frac{x}{2}, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.6 Fourier method

3.31 $

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