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Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+2 \cdot \frac{\partial u}{\partial z}=0, \quad u=y z \text { when } x=1 . \]

11.4.23 First order partial differential equations

3.05 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ x \frac{\partial z}{\partial x}-y \frac{\partial z}{\partial y}=0, \quad z=2 x \text { when } y=1 . \]

11.4.24 First order partial differential equations

2.54 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ \frac{\partial z}{\partial x}+\left(2 e^{x}-y\right) \frac{\partial z}{\partial y}=0, \quad z=y \text { when } x=0 . \]

11.4.25 First order partial differential equations

3.81 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}+x y \frac{\partial u}{\partial z}=0, \quad u=x^{2}+y^{2} \text { when } z=0 . \]

11.4.26 First order partial differential equations

3.81 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ x y \frac{\partial z}{\partial x}+x z \frac{\partial z}{\partial y}=y z, \quad z=1+y^{2} \text { when } x=1 . \]

11.4.27 First order partial differential equations

3.81 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ y \frac{\partial z}{\partial x}+x z \frac{\partial z}{\partial y}=y z, \quad z=-y^{2} \text { when } x=0 . \]

11.4.28 First order partial differential equations

3.05 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=x+y+z, z=x+y \text { when } y=x+1 . \]

11.4.29 First order partial differential equations

3.81 $

Problem: Find the surface, satisfying the equation \[ x z \frac{\partial z}{\partial x}+y z \frac{\partial z}{\partial y}=-x y \text {, } \] and passing through the line \( y=x^{2}, z=x^{3} \).

11.4.30 First order partial differential equations

3.81 $

Problem: Solve the first-order partial differential equation under the given additional conditions: \[ \left(y+2 u^{2}\right) \frac{\partial u}{\partial x}-2 x^{2} \cdot u \cdot \frac{\partial u}{\partial y}=x^{2}, x=u, \quad y=x^{2} . \]

11.4.31 First order partial differential equations

3.81 $

Problem: Find the general solution of the first-order partial differential equation: \[ e^{x} \frac{\partial u}{\partial x}+y^{2} \frac{\partial u}{\partial y}=y e^{x} \text {. } \]

11.4.32 First order partial differential equations

3.81 $

Problem: Find the solution of the first-order equation: \[ x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=2 x y \sqrt{a^{2}-z^{2}}, \quad a=\text { const } . \]

11.4.33 First order partial differential equations

3.81 $

Problem: Find the general solution of the first-order linear homogeneous equation: \[ (x-y+2) \frac{\partial u}{\partial x}+(2 x+3 y-1) \frac{\partial u}{\partial y}=0 . \]

11.4.6 First order partial differential equations

3.81 $

Problem: Find the solution of the homogeneous equation with the given initial and boundary conditions:

11.5.2.25 Fourier method

6.35 $

Problem: Find the solution of the Cauchy problem for the heatconduction equation: \[ \begin{array}{l} \frac{\partial u}{\partial t}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0,0 \leq x \leq 1, u(0, t)=u(1, t)=0, \\ u(x, 0)=f(x)=\left\{\begin{array}{ll} x, & 0 \leq x \leq 0.5 \\ 1-x, & 0.5 \leq x \leq 1 \end{array}\right. \end{array} \]

11.5.2.26 Fourier method

5.85 $

Problem: Find the solution of the Cauchy problem for the heatconduction equation: \[ \left\{\begin{array}{l} \frac{\partial T}{\partial t}=a^{2} \frac{\partial^{2} T}{\partial x^{2}}, \quad(0 \leq x \leq 4) \\ \left.T\right|_{x=0}=T(0, t)=0 \\ \left.T\right|_{x=4}=T(4, t)=0 \\ T(x, 0)=f(x)=\left\{\begin{array}{ll} 0, & 0 \leq x<1 \\ 1, & 1 \leq x \leq 4 \end{array}\right. \end{array}\right. \]

11.5.2.27 Fourier method

6.35 $

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