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Problem: Find the general solution of the equation, bringing it to canonical form: \[ 48 u_{x x}+16 u_{x y}+u_{y y}=0 . \]

11.5.4.17 With constant coefficients

2.61 $

Problem: Find the general solution of the equation, bringing it to canonical form: \[ 49 u_{x x}+28 u_{x y}+3 u_{y y}=0 . \]

11.5.4.18 With constant coefficients

3.39 $

Problem: Find the general solution of the equation, bringing it to canonical form: \[ u_{x x}-4 u_{x y}+4 u_{y y}+3 u_{x}-6 u_{y}=0 . \]

11.5.4.19 With constant coefficients

2.61 $

Problem: Find the particular solution of the second order partial differential equation satisfying the given conditions: \[ \begin{array}{l} z_{x x}+2 z_{x y}-3 z_{y y}+z_{\mathrm{x}}-z_{y}=0, \\ z(0, y)=y, \quad z_{\mathrm{x}}(0, y)=0 . \end{array} \]

11.5.4.20 With constant coefficients

3.91 $

Problem: - Find the general formula for solving the problem with given data on the curve; - define and depict on the plane the area in which the classical solution exists and is unique for any initial data: - if the data is given on both branches of the hyperbola, - if the data is given on the branch of the hyperbola, indicated in the table, - if the data is specified on the fragment of this branch, specified in the table; - find the solution for the initial data, given in the table; - determine whether this solution continues beyond the found areas and is unique there. \[ \begin{array}{l} u_{x y}-u_{y y}=0 .\left\{\begin{array}{l} \left.u\right|_{\Gamma}=\varphi(x) \\ \left.u_{x}\right|_{\Gamma}=\psi(x) \end{array} \quad \Gamma: x(x+y)=-1,\right. \\ \left\{\begin{array}{ll} \varphi(x)=x-1 & \text { the lower branch } \\ \psi(x)=-1 & \text { the fragment } x \in[3,4] \end{array}\right. \\ \end{array} \]

11.5.4.21 With constant coefficients

6.52 $

Problem: Bring the second order linear partial differential equation to canonical form. \[ \frac{\theta^{2} u}{\theta x^{2}}+2 \frac{\theta^{2} u}{\theta x \theta y}+4 \frac{\theta^{2} u}{\theta y^{2}}=-2 \frac{\theta u}{\theta x}-3 \frac{\theta u}{\theta y} . \]

11.5.4.22 With constant coefficients

2.61 $

Problem: Bring the second order linear partial differential equation to canonical form. \[ \frac{\theta^{2} u}{\theta x^{2}}-2 \frac{\theta^{2} u}{\theta x \theta y}+\frac{\theta^{2} u}{\theta y^{2}}+\frac{\theta u}{\theta x}+\frac{\theta u}{\theta y}+u=0 . \]

11.5.4.23 With constant coefficients

2.09 $

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

11.5.4.24 With constant coefficients

2.61 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{36} u_{x x}+17 \cos 4 t \cdot \sin 6 x \\ u(x, 0)=5 \sin 18 x, \quad u(0, t)=u(\pi, t)=0 . \end{array} \]

11.5.5.6 Mixed problems

7.83 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{25} u_{x x}+17 \sin 4 t \sin 5 x \\ u(x, 0)=5 \sin 15 x+3 \pi-2 x, u(0, t)=3 \pi, u(\pi, t)=\pi \end{array} \]

11.5.5.7 Mixed problems

7.83 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t t}=16 u_{x x}, u(0, t)=-4, u(1, t)=-1, \\ u(x, 0)=5 \sin 2 \pi x-4+3 x, \quad u_{t}(x, 0)=0 . \end{array} \]

11.5.5.8 Mixed problems

5.22 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t t}=\frac{1}{16} u_{x x}+50 e^{-7 t} \cdot \sin 4 x, \\ u(x, 0)=u_{t}(x, 0)=0, \quad u(0, t)=u(\pi, t)=0 . \end{array} \]

11.5.5.9 Mixed problems

6.52 $

Problem: Find the solution of the mixed problem: \[ \left\{\begin{array}{l} u_{t}=u_{x x}+u-x+2 \sin 2 x \cos x, \quad u=u(x, t) \\ 0

11.5.5.10 Mixed problems

10.44 $

Problem: Solve the problem of longitudinal oscillations of a rod suspended by the end \( \mathrm{x}=0 \) (the end \( x=l \) is free), performed under the influence of gravity.

11.5.5.11 Mixed problems

9.13 $

Problem: A thin homogeneous rod is given \( 0

11.5.5.12 Mixed problems

9.13 $

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