First order partial differential equations

MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

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

3.85 $

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

3.85 $

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

3.85 $

Problem: Find the surface that satisfies the first-order partial differential equation under the given conditions: \[ x \frac{\partial z}{\partial x}+z \frac{\partial z}{\partial y}=y, \quad y=2 z, \quad x+2 y=z . \]

11.4.34 First order partial differential equations

3.08 $

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

11.4.35 First order partial differential equations

3.85 $

Problem: Solve the Cauchy problem for the first-order partial differential equation: \[ x \frac{\partial u}{\partial x}+\left(y+x^{2}\right) \frac{\partial u}{\partial y}=u, \quad y=x, \quad u=x^{2} . \]

11.4.36 First order partial differential equations

3.85 $

Problem: Find the general integral of the first-order partial differential equation: \[ (x y-u) \frac{\partial u}{\partial x}+\left(1-y^{2}\right) \frac{\partial u}{\partial y}=y u-x . \]

11.4.37 First order partial differential equations

4.62 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ x y^{2} \frac{\partial u}{\partial x}+2 y^{3} \frac{\partial u}{\partial y}=2\left(y u-x^{2}\right), \quad y=1, \quad u=2 x^{2} . \]

11.4.38 First order partial differential equations

3.85 $

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

11.4.39 First order partial differential equations

3.34 $

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

11.4.40 First order partial differential equations

4.62 $

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

11.4.41 First order partial differential equations

3.85 $

Problem: Find the surface that satisfies the first-order partial differential equation under the given conditions: \[ x \frac{\partial z}{\partial x}-y \frac{\partial z}{\partial y}=z^{2}(x-3 y), x=1, \quad y z+1=0 . \]

11.4.42 First order partial differential equations

3.34 $

Problem: Find the surface that satisfies the first-order partial differential equation under the given conditions: \[ (y-z) \frac{\partial z}{\partial x}+(z-x) \frac{\partial z}{\partial y}=x-y, \quad z=y=-x . \]

11.4.43 First order partial differential equations

3.85 $

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

11.4.44 First order partial differential equations

3.85 $

‹
1
2
3