With constant coefficients

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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

6.37 $

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

4.33 $

Problem: Find all the values of \( \beta \), for which there is a linear transformation of variables \( (x, y) \rightarrow(z, t) \), bringing the equation to the following canonical form: a) \( u_{t t}= \) \( =u_{z z} \), b) \( u_{t}=u_{z z} \). \[ u_{x x}+4 u_{x y}-\beta u_{y y}-\beta u_{x}+\beta^{2} u_{y}=0 . \]

11.5.4.4 With constant coefficients

6.37 $

Problem: Determine the type of the second order equation: \[ u_{x x}+2 u_{x y}-2 u_{y z}+2 u_{y y}+2 u_{z z}=0 . \]

11.5.4.5 With constant coefficients

0 $

Problem: Determine the type of the second order equation: \[ u_{x y}+u_{x z}+u_{x t}+u_{z t}=0 . \]

11.5.4.6 With constant coefficients

1.53 $

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

2.55 $

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

2.55 $

Problem: Find the general solution of the equation, bringing it to canonical form: \[ u_{x x}+12 u_{x y}+36 u_{y y}+u_{x}+6 u_{y}=0 . \]

11.5.4.13 With constant coefficients

2.55 $

Problem: Determine the type of the second order equation. \[ 4 u_{x x}-4 u_{x y}-2 u_{y z}+u_{y}+u_{z}=0 . \]

11.5.4.14 With constant coefficients

1.27 $

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

11.5.4.15 With constant coefficients

2.55 $

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

11.5.4.16 With constant coefficients

2.55 $

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