6.2.4 2.4

   6.2.4.1 [470] problem number 1
   6.2.4.2 [471] problem number 2
   6.2.4.3 [472] problem number 3
   6.2.4.4 [473] problem number 4
   6.2.4.5 [474] problem number 5
   6.2.4.6 [475] problem number 6
   6.2.4.7 [476] problem number 7

6.2.4.1 [470] problem number 1

problem number 470

Added January 2, 2019.

Problem 2.2.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +(a \sqrt {x} y) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (y e^{-\frac {2}{3} a x^{3/2}}\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\frac {2 a x^{\frac {3}{2}}}{3}}\right )\]

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6.2.4.2 [471] problem number 2

problem number 471

Added January 2, 2019.

Problem 2.2.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +(a \sqrt {x} y+ b \sqrt {y}) w_y = 0 \]

Mathematica


\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {b \operatorname {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )}{\sqrt [3]{3} a^{2/3}}-\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\\& \left \{w(x,y)\to c_1\left (\frac {b \operatorname {Gamma}\left (\frac {2}{3},\frac {1}{3} a x^{3/2}\right )}{\sqrt [3]{3} a^{2/3}}+\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\\ \end {align*}

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (3 3^{\frac {1}{3}} b x \WhittakerM \left (\frac {1}{3}, \frac {5}{6}, \frac {a x^{\frac {3}{2}}}{3}\right ) {\mathrm e}^{\frac {a x^{\frac {3}{2}}}{6}}+5 \left (a x^{\frac {3}{2}}\right )^{\frac {1}{3}} b x -10 \left (a x^{\frac {3}{2}}\right )^{\frac {1}{3}} \sqrt {y}\right ) {\mathrm e}^{-\frac {a x^{\frac {3}{2}}}{3}}}{10 \left (a x^{\frac {3}{2}}\right )^{\frac {1}{3}}}\right )\]

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6.2.4.3 [472] problem number 3

problem number 472

Added January 2, 2019.

Problem 2.2.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +(a \sqrt {x} y+ b x \sqrt {y}) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt [3]{3} b \operatorname {Gamma}\left (\frac {4}{3},\frac {1}{3} a x^{3/2}\right )}{a^{4/3}}+\sqrt {y} e^{-\frac {1}{3} a x^{3/2}}\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (3 3^{\frac {1}{6}} b \sqrt {x}\, \WhittakerM \left (\frac {1}{6}, \frac {2}{3}, \frac {a x^{\frac {3}{2}}}{3}\right ) {\mathrm e}^{\frac {a x^{\frac {3}{2}}}{6}}-4 \left (a x^{\frac {3}{2}}\right )^{\frac {1}{6}} a \sqrt {y}\right ) {\mathrm e}^{-\frac {a x^{\frac {3}{2}}}{3}}}{4 \left (a x^{\frac {3}{2}}\right )^{\frac {1}{6}} a}\right )\]

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6.2.4.4 [473] problem number 4

problem number 473

Added January 2, 2019.

Problem 2.2.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x +A \sqrt {a x + b y+ c} w_y = 0 \]

Mathematica


\begin {align*} & \left \{w(x,y)\to c_1\left (x-\frac {a \log \left (\frac {e^{\frac {2 \sqrt {A^2 b^2 (a x+b y+c)}}{a}}}{\left (\sqrt {A^2 b^2 (a x+b y+c)}+a\right )^2}\right )}{A^2 b^2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (x-\frac {a \log \left (\frac {e^{-\frac {2 \sqrt {A^2 b^2 (a x+b y+c)}}{a}}}{\left (a-\sqrt {A^2 b^2 (a x+b y+c)}\right )^2}\right )}{A^2 b^2}\right )\right \}\\ \end {align*}

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {A^{2} b^{2} x -2 \sqrt {a x +b y +c}\, A b -a \ln \left (\sqrt {a x +b y +c}\, A b -a \right )+a \ln \left (\sqrt {a x +b y +c}\, A b +a \right )+a \ln \left (\left (a x +b y +c \right ) A^{2} b^{2}-a^{2}\right )}{A^{2} b^{2}}\right )\]

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6.2.4.5 [474] problem number 5

problem number 474

Added January 2, 2019.

Problem 2.2.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left ( a y + b \sqrt {y^2+c x^2} \right ) w_y = 0 \]

Mathematica


Failed

Maple


time expired

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6.2.4.6 [475] problem number 6

problem number 475

Added January 2, 2019.

Problem 2.2.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \left (a x + b \sqrt {y} \right ) w_x - \left ( c \sqrt {x} + a y \right ) w_y = 0 \]

Mathematica


\begin {align*} & \left \{w(x,y)\to c_1\left (\frac {3 a^3 x^3}{8 b^2}+\frac {2}{3} c x^{3/2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (a x y-\frac {2}{3} b y^{3/2}+\frac {2}{3} c x^{3/2}\right )\right \}\\& \left \{w(x,y)\to c_1\left (a x y+\frac {2}{3} b y^{3/2}+\frac {2}{3} c x^{3/2}\right )\right \}\\ \end {align*}

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\RootOf \left (3 a^{4} y^{4}+8 a b c^{2} y^{\frac {5}{2}}-2 \left (-a^{3} y^{3}-4 b c^{2} y^{\frac {3}{2}}-6 \mathit {\_Z} c^{2}+2 \sqrt {2 a^{3} b y^{\frac {9}{2}}+3 \mathit {\_Z} a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \mathit {\_Z} b c^{2} y^{\frac {3}{2}}+9 \mathit {\_Z}^{2} c^{2}}\, c \right )^{\frac {1}{3}} a^{3} y^{3}+12 \mathit {\_Z} a c^{2} y +3 \left (-a^{3} y^{3}-4 b c^{2} y^{\frac {3}{2}}-6 \mathit {\_Z} c^{2}+2 \sqrt {2 a^{3} b y^{\frac {9}{2}}+3 \mathit {\_Z} a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \mathit {\_Z} b c^{2} y^{\frac {3}{2}}+9 \mathit {\_Z}^{2} c^{2}}\, c \right )^{\frac {2}{3}} a^{2} y^{2}-4 \sqrt {2 a^{3} b y^{\frac {9}{2}}+3 \mathit {\_Z} a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \mathit {\_Z} b c^{2} y^{\frac {3}{2}}+9 \mathit {\_Z}^{2} c^{2}}\, a c y -4 \left (-a^{3} y^{3}-4 b c^{2} y^{\frac {3}{2}}-6 \mathit {\_Z} c^{2}+2 \sqrt {2 a^{3} b y^{\frac {9}{2}}+3 \mathit {\_Z} a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \mathit {\_Z} b c^{2} y^{\frac {3}{2}}+9 \mathit {\_Z}^{2} c^{2}}\, c \right )^{\frac {2}{3}} c^{2} x +\left (-a^{3} y^{3}-4 b c^{2} y^{\frac {3}{2}}-6 \mathit {\_Z} c^{2}+2 \sqrt {2 a^{3} b y^{\frac {9}{2}}+3 \mathit {\_Z} a^{3} y^{3}+4 b^{2} c^{2} y^{3}+12 \mathit {\_Z} b c^{2} y^{\frac {3}{2}}+9 \mathit {\_Z}^{2} c^{2}}\, c \right )^{\frac {4}{3}}\right )\right )\]

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6.2.4.7 [476] problem number 7

problem number 476

Added January 2, 2019.

Problem 2.2.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \sqrt {f(x)} w_x + \sqrt {f(y)} w_y = 0 \] Where \(f(t) = \sum _{n=0}^{4} a_n t^n \)

Mathematica


Failed

Maple


\[\text {Expression too large to display}\]

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