6.2.3 2.3

   6.2.3.1 [461] problem number 1
   6.2.3.2 [462] problem number 2
   6.2.3.3 [463] problem number 3
   6.2.3.4 [464] problem number 4
   6.2.3.5 [465] problem number 5
   6.2.3.6 [466] problem number 6
   6.2.3.7 [467] problem number 7
   6.2.3.8 [468] problem number 8
   6.2.3.9 [469] problem number 9

6.2.3.1 [461] problem number 1

problem number 461

Added January 2, 2019.

Problem 2.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(y^2+b x^2 y-a^2-a b x^2)w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (-\frac {e^{\frac {b x^3}{3}} \left (\left (b x^2+y\right ) \text {HeunT}\left [a^2,-((a-2) b),0,0,b,x\right ]+\text {HeunTPrime}\left [a^2,-((a-2) b),0,0,b,x\right ]\right )}{y \text {HeunT}\left [a^2,-a b,0,0,-b,x\right ]+\text {HeunTPrime}\left [a^2,-a b,0,0,-b,x\right ]}\right )\right \}\right \}\] But it can’t solve it when assuming \(b>0\) which is strange.

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {3 \left (\left (b x^{2} \mathrm {csgn}(b )-b x^{2}-2 y \right ) \mathit {HT}\left (-\frac {3^{\frac {2}{3}} a^{2}}{\left (b^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a -1\right ) \sqrt {b^{2}}}{b}, 0, \frac {3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} x}{3}\right )-\frac {2 3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} \mathit {HT}^{\prime }\left (-\frac {3^{\frac {2}{3}} a^{2}}{\left (b^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a -1\right ) \sqrt {b^{2}}}{b}, 0, \frac {3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} x}{3}\right )}{3}\right ) \mathit {HT}\left (-\frac {3^{\frac {2}{3}} a^{2}}{\left (b^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a -1\right ) \sqrt {b^{2}}}{b}, 0, \frac {3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} x}{3}\right )}{-3 \left (\left (b x^{2} \mathrm {csgn}(b )-b x^{2}-2 y \right ) \mathit {HT}\left (-\frac {3^{\frac {2}{3}} a^{2}}{\left (b^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a -1\right ) \sqrt {b^{2}}}{b}, 0, \frac {3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} x}{3}\right )-\frac {2 3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} \mathit {HT}^{\prime }\left (-\frac {3^{\frac {2}{3}} a^{2}}{\left (b^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a -1\right ) \sqrt {b^{2}}}{b}, 0, \frac {3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} x}{3}\right )}{3}\right ) \mathit {HT}\left (-\frac {3^{\frac {2}{3}} a^{2}}{\left (b^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a -1\right ) \sqrt {b^{2}}}{b}, 0, \frac {3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} x}{3}\right ) \left (\int \frac {{\mathrm e}^{\frac {\sqrt {b^{2}}\, x^{3}}{3}}}{\mathit {HT}\left (-\frac {3^{\frac {2}{3}} a^{2}}{\left (b^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a -1\right ) \sqrt {b^{2}}}{b}, 0, \frac {3^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{6}} x}{3}\right )^{2}}d x \right )+6 \,{\mathrm e}^{\frac {\sqrt {b^{2}}\, x^{3}}{3}}}\right )\]

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6.2.3.2 [462] problem number 2

problem number 462

Added January 2, 2019.

Problem 2.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +(a x^2 y+b x^3+c) w_y = 0 \]

Mathematica


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

Maple


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

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6.2.3.3 [463] problem number 3

problem number 463

Added January 2, 2019.

Problem 2.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +(a x^2 y+b y^3) w_y = 0 \]

Mathematica


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

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {-2 2^{\frac {2}{3}} 3^{\frac {5}{6}} O b x y^{2}+3 2^{\frac {2}{3}} 3^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right ) b x y^{2} \Gamma \left (\frac {1}{3}, -\frac {2 a x^{3}}{3}\right )-9 \Gamma \left (\frac {2}{3}\right ) O \,{\mathrm e}^{\frac {2 a x^{3}}{3}}}{9 \Gamma \left (\frac {2}{3}\right ) O y^{2}}\right )\]

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6.2.3.4 [464] problem number 4

problem number 464

Added January 2, 2019.

Problem 2.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x +(a x y+b) y^2 w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-2 \sqrt {b^{2}-4 a}\, b \arctanh \left (\frac {\sqrt {b^{2}-4 a}\, \left (2 a y x +b \right )}{-b^{2}+4 a}\right )-8 \left (\ln \left (x y \right )-\frac {\ln \left (\left (a y^{2} x^{2}+b y x +1\right ) x^{2}\right )}{2}\right ) \left (-\frac {b^{2}}{4}+a \right )}{-2 b^{2}+8 a}\right )\]

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6.2.3.5 [465] problem number 5

problem number 465

Added January 2, 2019.

Problem 2.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +A(a x+b y+c)^3 y^2 w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (x -\frac {\ln \left (\frac {a x +\left (-\RootOf \left (A b^{4} \mathit {\_Z}^{3}+3 A b^{3} c \mathit {\_Z}^{2}+3 A b^{2} c^{2} \mathit {\_Z} +A b c^{3}+a \right )+y \right ) b}{b}\right )}{3 A b \left (\RootOf \left (A b^{4} \mathit {\_Z}^{3}+3 A b^{3} c \mathit {\_Z}^{2}+3 A b^{2} c^{2} \mathit {\_Z} +A b c^{3}+a \right ) b +c \right )^{2}}\right )\] Answer contains RootOf

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6.2.3.6 [466] problem number 6

problem number 466

Added January 2, 2019.

Problem 2.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x +(a x^4 y^3+(b x^2-1)y+c x) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {b^{3} \ln \left (\frac {-b y x -\RootOf \left (c^{2} a \mathit {\_Z}^{3}+b^{3} \mathit {\_Z} -b^{3}\right ) c}{c}\right )}{3 \RootOf \left (c^{2} a \mathit {\_Z}^{3}+b^{3} \mathit {\_Z} -b^{3}\right )^{2} a c^{2}+b^{3}}-\frac {b x^{2}}{2}\right )\] Answer contains RootOf

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6.2.3.7 [467] problem number 7

problem number 467

Added January 2, 2019.

Problem 2.2.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x +(a x^2 y^2+b x y+c) w_y = 0 \]

Mathematica


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

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-2 \arctan \left (\frac {2 a y x +b +1}{\sqrt {4 a c -b^{2}-2 b -1}}\right )+\sqrt {4 a c -b^{2}-2 b -1}\, \ln (x )}{\sqrt {4 a c -b^{2}-2 b -1}}\right )\]

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6.2.3.8 [468] problem number 8

problem number 468

Added January 2, 2019.

Problem 2.2.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^2 y+b) w_x -(a x y^2+c) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\frac {a x^2 y^2+2 b y+2 c x}{a}\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {1}{2} a x^{2} y^{2}-b y -c x \right )\]

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6.2.3.9 [469] problem number 9

problem number 469

Added January 2, 2019.

Problem 2.2.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x+b y^3) w_x -(c x^3+a y) w_y = 0 \]

Mathematica


$Aborted

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {1}{4} b y^{4}-\frac {1}{4} c x^{4}-a x y \right )\]

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