6.2.5 2.5

   6.2.5.1 [477] problem number 1
   6.2.5.2 [478] problem number 2
   6.2.5.3 [479] problem number 3
   6.2.5.4 [480] problem number 4
   6.2.5.5 [481] problem number 5
   6.2.5.6 [482] problem number 6
   6.2.5.7 [483] problem number 7
   6.2.5.8 [484] problem number 8
   6.2.5.9 [485] problem number 9
   6.2.5.10 [486] problem number 10
   6.2.5.11 [487] problem number 11
   6.2.5.12 [488] problem number 12
   6.2.5.13 [489] problem number 13
   6.2.5.14 [490] problem number 14
   6.2.5.15 [491] problem number 15
   6.2.5.16 [492] problem number 16
   6.2.5.17 [493] problem number 17
   6.2.5.18 [494] problem number 18
   6.2.5.19 [495] problem number 19
   6.2.5.20 [496] problem number 20
   6.2.5.21 [497] problem number 21
   6.2.5.22 [498] problem number 22
   6.2.5.23 [499] problem number 23
   6.2.5.24 [500] problem number 24
   6.2.5.25 [501] problem number 25
   6.2.5.26 [502] problem number 26
   6.2.5.27 [503] problem number 27
   6.2.5.28 [504] problem number 28
   6.2.5.29 [505] problem number 29
   6.2.5.30 [506] problem number 30
   6.2.5.31 [507] problem number 31
   6.2.5.32 [508] problem number 32
   6.2.5.33 [509] problem number 33
   6.2.5.34 [510] problem number 34
   6.2.5.35 [511] problem number 35
   6.2.5.36 [512] problem number 36
   6.2.5.37 [513] problem number 37
   6.2.5.38 [514] problem number 38
   6.2.5.39 [515] problem number 39
   6.2.5.40 [516] problem number 40
   6.2.5.41 [517] problem number 41
   6.2.5.42 [518] problem number 42
   6.2.5.43 [519] problem number 43
   6.2.5.44 [520] problem number 44
   6.2.5.45 [521] problem number 45
   6.2.5.46 [522] problem number 46
   6.2.5.47 [523] problem number 47
   6.2.5.48 [524] problem number 48
   6.2.5.49 [525] problem number 49
   6.2.5.50 [526] problem number 50
   6.2.5.51 [527] problem number 51
   6.2.5.52 [528] problem number 52
   6.2.5.53 [529] problem number 53
   6.2.5.54 [530] problem number 54
   6.2.5.55 [531] problem number 55
   6.2.5.56 [532] problem number 56

6.2.5.1 [477] problem number 1

problem number 477

Added January 2, 2019.

Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (b a^{-k-1} \operatorname {Gamma}(k+1,a x)+y e^{-a x}\right )\right \}\right \}\]

Maple


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

________________________________________________________________________________________

6.2.5.2 [478] problem number 2

problem number 478

Added January 2, 2019.

Problem 2.2.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^k y+b x^n \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (b (k+1)^{\frac {n-k}{k+1}} a^{-\frac {n+1}{k+1}} \operatorname {Gamma}\left (\frac {n+1}{k+1},\frac {a x^{k+1}}{k+1}\right )+y e^{-\frac {a x^{k+1}}{k+1}}\right )\right \}\right \}\]

Maple


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

________________________________________________________________________________________

6.2.5.3 [479] problem number 3

problem number 479

Added January 2, 2019.

Problem 2.2.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a x y \BesselY \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x x^{\frac {n}{2}}}{n +2}\right )-\sqrt {a b}\, x x^{\frac {n}{2}} \BesselY \left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x x^{\frac {n}{2}}}{n +2}\right )+\BesselY \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x x^{\frac {n}{2}}}{n +2}\right )}{-a x y \BesselJ \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x x^{\frac {n}{2}}}{n +2}\right )+\sqrt {a b}\, x x^{\frac {n}{2}} \BesselJ \left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x x^{\frac {n}{2}}}{n +2}\right )-\BesselJ \left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x x^{\frac {n}{2}}}{n +2}\right )}\right )\]

________________________________________________________________________________________

6.2.5.4 [480] problem number 4

problem number 480

Added January 2, 2019.

Problem 2.2.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( y^2+a n x^{n-1} -a^2 x^{2 n} \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {2 \left (a x^{\frac {5 n}{2}+2}-y x^{\frac {3 n}{2}+2}\right ) {\mathrm e}^{-\frac {2 a x^{n +1}}{n +1}}}{-3 \left (\left (n +\frac {4}{3}\right ) a x^{n +1}+\frac {\left (n +2\right ) \left (-x y +n +1\right )}{3}\right ) \left (n +2\right ) \WhittakerM \left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x^{n +1}}{n +1}\right ) {\mathrm e}^{-\frac {a x^{n +1}}{n +1}}+2 \left (a^{2} x^{2 n +2}-a y x^{n +2}-\frac {\left (n +2\right ) a x^{n +1}}{2}-\frac {\left (n +2\right ) \left (-x y +n +1\right )}{2}\right ) \left (n +1\right ) \WhittakerM \left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x^{n +1}}{n +1}\right ) {\mathrm e}^{-\frac {a x^{n +1}}{n +1}}+2 \left (n +2\right )^{2} \left (n +\frac {3}{2}\right ) \left (-\frac {2 a x^{n +1}}{n +1}\right )^{\frac {3 n +4}{2 n +2}}}\right )\]

________________________________________________________________________________________

6.2.5.5 [481] problem number 5

problem number 481

Added January 2, 2019.

Problem 2.2.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( y^2 + a x^n y + a x^{n-1} \right ) w_y = 0 \]

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.6 [482] problem number 6

problem number 482

Added January 2, 2019.

Problem 2.2.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (-b +y \right ) \left (\int {\mathrm e}^{\frac {\left (a x^{n}+2 \left (n +1\right ) b \right ) x}{n +1}}d x \right )+{\mathrm e}^{\frac {\left (a x^{n}+2 \left (n +1\right ) b \right ) x}{n +1}}}{b -y}\right )\]

________________________________________________________________________________________

6.2.5.7 [483] problem number 7

problem number 483

Added January 2, 2019.

Problem 2.2.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.8 [484] problem number 8

problem number 484

Added January 2, 2019.

Problem 2.2.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2 + b m x^{m-1} -a b^2 x^{n+2 m} \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (-b x^{\frac {5 m}{2}+2 n +2}+y x^{\frac {3 m}{2}+2 n +2}\right ) a \,{\mathrm e}^{-\frac {2 a b x^{m +n +1}}{m +n +1}}}{-3 \left (\left (m +\frac {4 n}{3}+\frac {4}{3}\right ) a b x^{m +n +1}-\frac {\left (m +2 n +2\right ) \left (a y x^{n +1}-m -n -1\right )}{3}\right ) \left (m +2 n +2\right ) \WhittakerM \left (\frac {m +2 n +2}{2 m +2 n +2}, \frac {2 m +3 n +3}{2 m +2 n +2}, -\frac {2 a b x^{m +n +1}}{m +n +1}\right ) {\mathrm e}^{-\frac {a b x^{m +n +1}}{m +n +1}}+2 \left (a^{2} b^{2} x^{2 m +2 n +2}-a^{2} b y x^{m +2 n +2}-\frac {\left (m +2 n +2\right ) \left (a b x^{m +n +1}-a y x^{n +1}+m +n +1\right )}{2}\right ) \left (m +n +1\right ) \WhittakerM \left (-\frac {m}{2 m +2 n +2}, \frac {2 m +3 n +3}{2 m +2 n +2}, -\frac {2 a b x^{m +n +1}}{m +n +1}\right ) {\mathrm e}^{-\frac {a b x^{m +n +1}}{m +n +1}}+2 \left (m +\frac {3 n}{2}+\frac {3}{2}\right ) \left (m +2 n +2\right )^{2} \left (-\frac {2 a b x^{m +n +1}}{m +n +1}\right )^{\frac {3 m +4 n +4}{2 m +2 n +2}}}\right )\]

________________________________________________________________________________________

6.2.5.9 [485] problem number 9

problem number 485

Added January 2, 2019.

Problem 2.2.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( (n+1)x^n y^2 - a x^{n+m+1} y + a x^m \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (\left (m -n \right ) a x x^{m} \hypergeom \left (\left [\frac {2 m +2}{m +n +2}\right ], \left [\frac {2 m +n +3}{m +n +2}\right ], \frac {a x^{2} x^{m} x^{n}}{m +n +2}\right )-\left (m +1\right ) \left (a x x^{m}-\left (n +1\right ) y \right ) \hypergeom \left (\left [\frac {m -n}{m +n +2}\right ], \left [\frac {m +1}{m +n +2}\right ], \frac {a x^{2} x^{m} x^{n}}{m +n +2}\right )\right ) \left (m +2 n +3\right )}{\left (\left (m +1\right ) a x^{2} x^{m} x^{n} \hypergeom \left (\left [\frac {2 m +n +3}{m +n +2}\right ], \left [\frac {2 m +3 n +5}{m +n +2}\right ], \frac {a x^{2} x^{m} x^{n}}{m +n +2}\right )-\left (m +2 n +3\right ) \left (\left (a x x^{m}-\left (n +1\right ) y \right ) x x^{n}-n -1\right ) \hypergeom \left (\left [\frac {m +1}{m +n +2}\right ], \left [\frac {m +2 n +3}{m +n +2}\right ], \frac {a x^{2} x^{m} x^{n}}{m +n +2}\right )\right ) \left (m +1\right )}\right )\]

________________________________________________________________________________________

6.2.5.10 [486] problem number 10

problem number 486

Added January 2, 2019.

Problem 2.2.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (-c -y \right ) \left (\int a x^{n} {\mathrm e}^{\frac {-2 \left (m +1\right ) a c x^{n +1}+\left (n +1\right ) b x^{m +1}}{\left (m +1\right ) \left (n +1\right )}}d x \right )-{\mathrm e}^{\frac {-2 \left (m +1\right ) a c x^{n +1}+\left (n +1\right ) b x^{m +1}}{\left (m +1\right ) \left (n +1\right )}}}{c +y}\right )\]

________________________________________________________________________________________

6.2.5.11 [487] problem number 11

problem number 487

Added January 2, 2019.

Problem 2.2.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2-a x^n (b x^m +c) y+ b m x^{m-1} \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.12 [488] problem number 12

problem number 488

Added January 2, 2019.

Problem 2.2.5.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x - \left (a n x^{n-1} y^2 - c x^m (a x^n+b) + c x^m \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.13 [489] problem number 13

problem number 489

Added January 2, 2019.

Problem 2.2.5.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^2+b x^m y+ c k x^{k-1}-b c x^{m+k}-a c^2 x^{n+2 k} \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.14 [490] problem number 14

problem number 490

Added January 2, 2019.

Problem 2.2.5.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^{2 n+1} y^3 + b x^{-n-2} \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\ln (x )-\frac {\ln \left (x y x^{n}-\RootOf \left (\mathit {\_Z}^{3} a +\left (n +1\right ) \mathit {\_Z} +b \right )\right )}{3 \RootOf \left (\mathit {\_Z}^{3} a +\left (n +1\right ) \mathit {\_Z} +b \right )^{2} a +n +1}\right )\] Solution contains RootOf

________________________________________________________________________________________

6.2.5.15 [491] problem number 15

problem number 491

Added January 2, 2019.

Problem 2.2.5.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2 - b m x^{m-1} - 2 a b^3 x^{n+3 m} \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\frac {6^{-\frac {n+1}{2 m+n+1}} (2 m+n+1)^{-\frac {2 m}{2 m+n+1}} b^{-\frac {2 (n+1)}{2 m+n+1}} e^{-\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \left (6^{\frac {n+1}{2 m+n+1}} (2 m+n+1)^{\frac {2 m}{2 m+n+1}} b^{\frac {2 (n+1)}{2 m+n+1}}-2 a^{\frac {2 m}{2 m+n+1}} \left (b x^m+y\right )^2 e^{\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}} \operatorname {Gamma}\left (\frac {n+1}{2 m+n+1},\frac {6 a b^2 x^{2 m+n+1}}{2 m+n+1}\right )\right )}{\left (b x^m+y\right )^2}\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (\left (n +1\right ) \left (n^{2}-n +1\right ) \left (y^{2} x^{-2 m}+b^{2}\right ) 2^{\frac {m}{2 m +n +1}}+\left (4 \left (m^{3}+\left (\frac {5 n}{2}+\frac {5}{2}\right ) m^{2}+\frac {3 n^{2}}{2}+2 \left (n +1\right )^{2} m +\frac {3 n}{2}\right ) y^{2} x^{-2 m}+4 \left (2 \left (m +n +1\right )^{2} \left (m +\frac {n}{2}+\frac {1}{2}\right ) y x^{-m}+\left (m^{3}+\left (\frac {5 n}{2}+\frac {5}{2}\right ) m^{2}+\frac {3 n^{2}}{2}+2 \left (n +1\right )^{2} m +\frac {3 n}{2}\right ) b \right ) b \right ) 2^{\frac {-m -n -1}{2 m +n +1}}\right ) 3^{\frac {-3 m -2 n -2}{2 m +n +1}} \left (\frac {a b^{2} x^{2 m +n +1}}{2 m +n +1}\right )^{\frac {-m -n -1}{2 m +n +1}} \WhittakerM \left (\frac {m +n +1}{2 m +n +1}, \frac {4 m +3 n +3}{4 m +2 n +2}, \frac {6 a b^{2} x^{2 m +n +1}}{2 m +n +1}\right ) {\mathrm e}^{-\frac {3 a b^{2} x^{2 m +n +1}}{2 m +n +1}}+4 \left (n +1\right ) \left (m +n +1\right ) \left (m +\frac {3 n}{4}+\frac {3}{4}\right ) b^{2} {\mathrm e}^{-\frac {6 a b^{2} x^{2 m +n +1}}{2 m +n +1}}+\left (\left (\left (n^{2}+1\right ) a b^{4} x^{2 m +n +1}+\left (n +1\right ) n y^{2} x^{-2 m}+\left (\left (n^{2}+1\right ) a y^{2} x^{n +1}+n^{2}+n \right ) b^{2}\right ) 2^{\frac {-m -n -1}{2 m +n +1}} 3^{\frac {-m -n -1}{2 m +n +1}}+\left (\left (n +1\right ) \left (n^{2}-n +1\right ) b y 2^{\frac {m}{2 m +n +1}} x^{-m}+\left (12 \left (m^{2}+\left (n +1\right ) m +\frac {n}{2}\right ) a b^{4} x^{2 m +n +1}+24 \left (m +\frac {n}{2}+\frac {1}{2}\right )^{2} a b^{3} y x^{m +n +1}+4 \left (m^{3}+\frac {n^{3}}{4}+\left (2 n +2\right ) m^{2}+\frac {5 \left (n +1\right )^{2} m}{4}+\frac {1}{4}\right ) y^{2} x^{-2 m}+12 \left (\frac {2 \left (m^{3}+\left (2 n +2\right ) m^{2}+\frac {3 n^{2}}{4}+\frac {5 \left (n +1\right )^{2} m}{4}+\frac {3 n}{4}\right ) y x^{-m}}{3}+\left (\left (m^{2}+\left (n +1\right ) m +\frac {n}{2}\right ) a y^{2} x^{n +1}+\frac {m^{3}}{3}+\frac {n^{3}}{12}+\left (\frac {2 n}{3}+\frac {2}{3}\right ) m^{2}+\frac {5 \left (n +1\right )^{2} m}{12}+\frac {1}{12}\right ) b \right ) b \right ) 2^{\frac {-m -n -1}{2 m +n +1}}\right ) 3^{\frac {-3 m -2 n -2}{2 m +n +1}}\right ) \left (\frac {a b^{2} x^{2 m +n +1}}{2 m +n +1}\right )^{\frac {-m -n -1}{2 m +n +1}} \WhittakerM \left (-\frac {m}{2 m +n +1}, \frac {4 m +3 n +3}{4 m +2 n +2}, \frac {6 a b^{2} x^{2 m +n +1}}{2 m +n +1}\right ) {\mathrm e}^{-\frac {3 a b^{2} x^{2 m +n +1}}{2 m +n +1}}}{\left (n +1\right ) \left (m +n +1\right ) \left (4 m +3 n +3\right ) \left (b^{2} x^{2 m}+2 b y x^{m}+y^{2}\right ) b^{2}}\right )\]

________________________________________________________________________________________

6.2.5.16 [492] problem number 16

problem number 492

Added January 2, 2019.

Problem 2.2.5.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^3 + 3 a b x^{n+m} y^2+ c x^k y- 2 a b^3 x^{n+3 m} + b c x^{m+l} - b m x^{m-1} \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.17 [493] problem number 17

problem number 493

Added January 2, 2019.

Problem 2.2.5.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a y^n + b x ^{\frac {n}{1-n}} \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-n \left (\int _{\mathit {\_b}}^{y}\frac {x^{\frac {n}{n -1}}}{\left (n -1\right ) b x +\left (\left (n -1\right ) a x \mathit {\_a}^{n}+\mathit {\_a} \right ) x^{\frac {n}{n -1}}}d \mathit {\_a} \right )+\int _{\mathit {\_b}}^{y}\frac {x^{\frac {n}{n -1}}}{\left (n -1\right ) b x +\left (\left (n -1\right ) a x \mathit {\_a}^{n}+\mathit {\_a} \right ) x^{\frac {n}{n -1}}}d \mathit {\_a} +\ln (x )\right )\]

________________________________________________________________________________________

6.2.5.18 [494] problem number 18

problem number 494

Added January 2, 2019.

Problem 2.2.5.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^{m-n-(m n)} y^n + b x^m \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\int _{\mathit {\_b}}^{y}-\frac {x^{n} x^{m n}}{a x \mathit {\_a}^{n} x^{m}+\left (b x x^{m}-\left (m +1\right ) \mathit {\_a} \right ) x^{n} x^{m n}}d \mathit {\_a} +\ln (x )\right )\]

________________________________________________________________________________________

6.2.5.19 [495] problem number 19

problem number 495

Added January 2, 2019.

Problem 2.2.5.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( a x^n y^k + b x^m y \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (a (-1)^{\frac {m-n}{m+1}} (m+1)^{\frac {n-m}{m+1}} b^{-\frac {n+1}{m+1}} (k-1)^{\frac {m-n}{m+1}} \operatorname {Gamma}\left (\frac {n+1}{m+1},-\frac {b (k-1) x^{m+1}}{m+1}\right )+y^{1-k} e^{\frac {b (k-1) x^{m+1}}{m+1}}\right )\right \}\right \}\]

Maple


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

________________________________________________________________________________________

6.2.5.20 [496] problem number 20

problem number 496

Added January 2, 2019.

Problem 2.2.5.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.21 [497] problem number 21

problem number 497

Added January 2, 2019.

Problem 2.2.5.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( a y^2+(n+b x^n) y + c x^{2 n} \right ) w_y = 0 \]

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.22 [498] problem number 22

problem number 498

Added January 2, 2019.

Problem 2.2.5.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.23 [499] problem number 23

problem number 499

Added January 2, 2019.

Problem 2.2.5.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


Failed

Maple


Failed to convert to latex

________________________________________________________________________________________

6.2.5.24 [500] problem number 24

problem number 500

Added January 2, 2019.

Problem 2.2.5.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left ( x^{2 n} y^2+(m-n) y+ x^{2 m} \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (y x^{n-m}\right )-\frac {x^{m+n}}{m+n}\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {x^{m +n}+\left (-m -n \right ) \arctan \left (y x^{-m +n}\right )}{m +n}\right )\]

________________________________________________________________________________________

6.2.5.25 [501] problem number 25

problem number 501

Added January 2, 2019.

Problem 2.2.5.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.26 [502] problem number 26

problem number 502

Added January 2, 2019.

Problem 2.2.5.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.27 [503] problem number 27

problem number 503

Added January 2, 2019.

Problem 2.2.5.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left ( a y^3+3 a b x^n y^2 - b n x^n -2 a b^3 x^{3 n} \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\frac {e^{-\frac {3 a b^2 x^{2 n}}{n}} \left (a e^{\frac {3 a b^2 x^{2 n}}{n}} \left (b x^n+y\right )^2 \text {Ei}\left (-\frac {3 a b^2 x^{2 n}}{n}\right )+n\right )}{n \left (b x^n+y\right )^2}\right )\right \}\right \}\]

Maple


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

________________________________________________________________________________________

6.2.5.28 [504] problem number 28

problem number 504

Added January 2, 2019.

Problem 2.2.5.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


Failed

Maple


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

________________________________________________________________________________________

6.2.5.29 [505] problem number 29

problem number 505

Added January 2, 2019.

Problem 2.2.5.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


Failed

Maple


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

________________________________________________________________________________________

6.2.5.30 [506] problem number 30

problem number 506

Added January 2, 2019.

Problem 2.2.5.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + \left ( y+a x^{n - m }y^m+b x^{n-k} y^k \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (n -1\right ) x \left (\int _{\mathit {\_b}}^{y}-\frac {x^{k} x^{m}}{\left (a \mathit {\_a}^{m} x^{k}+b \mathit {\_a}^{k} x^{m}\right ) x}d \mathit {\_a} \right )+x^{n}}{\left (n -1\right ) x}\right )\]

________________________________________________________________________________________

6.2.5.31 [507] problem number 31

problem number 507

Added January 2, 2019.

Problem 2.2.5.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ y w_x + \left (x^{n-1}((1+2 n)x+a n) y-n x^{2 n}(x+a) \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (a +\frac {x}{2}\right ) n \,{\mathrm e}^{\frac {2 \arctan \left (\frac {\left (2 a x^{n}-y +x^{n +1}\right ) n}{\sqrt {-n^{2}}\, \left (-y +x^{n +1}\right )}\right )}{\sqrt {-n^{2}}}}-\sqrt {-n^{2}}\, x \left (\int _{}^{-\frac {2 \arctan \left (\frac {\left (2 a x^{n}-y +x^{n +1}\right ) n}{\sqrt {-n^{2}}\, \left (-y +x^{n +1}\right )}\right )}{\sqrt {-n^{2}}}}{\mathrm e}^{-\mathit {\_a}} \tan \left (\frac {\sqrt {-n^{2}}\, \mathit {\_a}}{2}\right )d\mathit {\_a} \right )}{2 x}\right )\]

________________________________________________________________________________________

6.2.5.32 [508] problem number 32

problem number 508

Added January 2, 2019.

Problem 2.2.5.32 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ y w_x + \left ( (a(2 n +k)x^k+b)x^{n-1}y -(a^2 n x^{2 k}+ a b x^k -c) x^{2 n-1} \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.33 [509] problem number 33

problem number 509

Added January 2, 2019.

Problem 2.2.5.33 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.34 [510] problem number 34

problem number 510

Added January 2, 2019.

Problem 2.2.5.34 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


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

Maple


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

________________________________________________________________________________________

6.2.5.35 [511] problem number 35

problem number 511

Added January 2, 2019.

Problem 2.2.5.35 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\frac {b x^{1-m}}{a (m-1)}-\frac {y^{1-n}}{n-1}\right )\right \}\right \}\]

Maple


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

________________________________________________________________________________________

6.2.5.36 [512] problem number 36

problem number 512

Added January 2, 2019.

Problem 2.2.5.36 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (y e^{\frac {b x^{1-n}}{a (n-1)}}-\frac {c (a-a n)^{\frac {-m+n-1}{n-1}} b^{\frac {m-n+1}{n-1}} \operatorname {Gamma}\left (\frac {-m+n-1}{n-1},\frac {b x^{1-n}}{a-a n}\right )}{a (n-1)}\right )\right \}\right \}\]

Maple


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

________________________________________________________________________________________

6.2.5.37 [513] problem number 37

problem number 513

Added January 2, 2019.

Problem 2.2.5.37 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left ((k-m-1)^{\frac {m}{k-m-1}} a^{\frac {m}{k-m-1}} b^{\frac {1-k}{k-m-1}} (n-1)^{\frac {m}{-k+m+1}} \operatorname {Gamma}\left (\frac {k-1}{k-m-1},\frac {b (n-1) x^{-k+m+1}}{a (k-m-1)}\right )+y^{1-n} e^{-\frac {b (n-1) x^{-k+m+1}}{a (k-m-1)}}\right )\right \}\right \}\]

Maple


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

________________________________________________________________________________________

6.2.5.38 [514] problem number 38

problem number 514

Added January 2, 2019.

Problem 2.2.5.38 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x(a x^k+b) w_x + \left ( \alpha x^n y^2+(\beta -a n x^k)y+\gamma x^{-n} \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+2 \alpha y x^n+b n+\beta \right ) \exp \left (\frac {\sqrt {\alpha } \sqrt {\gamma } \left (k \log (x)-\log \left (a x^k+b\right )\right ) \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}}{b k}\right )}{-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+2 \alpha y x^n+b n+\beta }\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {\left (2 \left (b n +\beta \right ) b k \arctanh \left (\frac {\left (b n +\beta \right ) \left (2 \alpha y x^{n}+b n +\beta \right )}{\sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 \alpha g +\beta ^{2}\right )}}\right )+\sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 \alpha g +\beta ^{2}\right )}\, \left (k \ln (x )-\ln \left (a x^{k}+b \right )\right )\right ) \left (b n +\beta \right )}{\sqrt {\left (b n +\beta \right )^{2} \left (b^{2} n^{2}+2 b \beta n -4 \alpha g +\beta ^{2}\right )}\, b k}\right )\]

________________________________________________________________________________________

6.2.5.39 [515] problem number 39

problem number 515

Added January 2, 2019.

Problem 2.2.5.39 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (y+ A x^n + a) w_x - \left ( n A x^{n-1} y + k x^m + b\right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (y \left (2 a+2 A x^n+y\right )+2 b x+\frac {2 k x^{m+1}}{m+1}\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-2 k x x^{m}-2 \left (m +1\right ) \left (A y x^{n}+a y +b x +\frac {y^{2}}{2}\right )}{2 m +2}\right )\]

________________________________________________________________________________________

6.2.5.40 [516] problem number 40

problem number 516

Added January 2, 2019.

Problem 2.2.5.40 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (y+ a x^{n+1}+b x^n) w_x + \left (a n x^n + c x^{n-1} \right ) y w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.41 [517] problem number 41

problem number 517

Added January 2, 2019.

Problem 2.2.5.41 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x(2 a x^n y+b) w_x - \left (a(3 n+m)x^n y^2+b(2 n+m)y-A x^m -C x^{-n} \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\frac {x^{m+n} \left (x^n \left (2 y (m+n) \left (a y x^n+b\right )-A x^m\right )-2 \text {C0}\right )}{2 a (m+n)}\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-2 \left (m +n \right ) a y^{2} x^{m +3 n}-2 \left (m +n \right ) b y x^{m +2 n}+A x^{2 m +2 n}+2 C x^{m +n}}{2 m +2 n}\right )\]

________________________________________________________________________________________

6.2.5.42 [518] problem number 42

problem number 518

Added January 2, 2019.

Problem 2.2.5.42 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (\left (-n +1\right ) \ln \left (\frac {9 \left (a^{2} x^{n}+\left (a b +c \right ) x^{2}\right ) \left (n^{2}-3 n +3\right )}{\left (n -3\right ) \left (a x^{n}+\left (b x +y \right ) x \right ) a}\right )+\ln \left (\frac {9 \left (\left (n -1\right ) a^{2} x^{n}+\left (c x +\left (-b x +\left (b x +y \right ) n -2 y \right ) a \right ) x \right ) \left (n^{2}-3 n +3\right )}{\left (2 n -3\right ) \left (b x^{2}+a x^{n}+x y \right ) a}\right )+\left (n -2\right ) \ln \left (-\frac {9 \left (n^{2}-3 n +3\right ) \left (a y -c x \right ) x}{\left (a x^{n}+\left (b x +y \right ) x \right ) a n}\right )-2 \left (\ln (x )-\frac {\ln \left (a^{2} x^{n}+\left (a b +c \right ) x^{2}\right )}{2}\right ) \left (n -1\right )\right ) \left (n^{2}-3 n +3\right )}{3 \left (n -2\right ) \left (n -1\right )}\right )\]

________________________________________________________________________________________

6.2.5.43 [519] problem number 43

problem number 519

Added January 2, 2019.

Problem 2.2.5.43 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (n^{2}-3 n +3\right ) \left (\left (-n +1\right ) \ln \left (\frac {9 \left (n^{2}-3 n +3\right ) \left (k^{2} y^{n}+\left (a b +c k \right ) y^{2}\right )}{\left (n -3\right ) \left (b x y +c y^{2}+k y^{n}\right ) k}\right )+\ln \left (\frac {9 \left (n^{2}-3 n +3\right ) \left (\left (n -1\right ) k^{2} y^{n}+\left (a b y +\left (\left (n -2\right ) b x +\left (n -1\right ) c y \right ) k \right ) y \right )}{2 \left (n -\frac {3}{2}\right ) \left (b x y +c y^{2}+k y^{n}\right ) k}\right )+\left (n -2\right ) \ln \left (\frac {9 \left (n^{2}-3 n +3\right ) \left (a y -x k \right ) b y}{\left (b x y +c y^{2}+k y^{n}\right ) k n}\right )+\left (n -1\right ) \left (-2 \ln (y )+\ln \left (a b y^{2}+c k y^{2}+k^{2} y^{n}\right )\right )\right )}{3 \left (n -2\right ) \left (n -1\right )}\right )\]

________________________________________________________________________________________

6.2.5.44 [520] problem number 44

problem number 520

Added January 2, 2019.

Problem 2.2.5.44 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left (c y^2-b x^{m-1} y+ a x^{n-2}\right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.45 [521] problem number 45

problem number 521

Added January 2, 2019.

Problem 2.2.5.45 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left (a x^{n-2} y^2 + b x^{m-1} y + c \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.46 [522] problem number 46

problem number 522

Added January 2, 2019.

Problem 2.2.5.46 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left ( \alpha x^k y^2 + \beta x^s y - \alpha \lambda ^2 x^k + \beta \lambda x^s \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-\left (\lambda +y \right ) \alpha \left (\int \frac {x^{k} {\mathrm e}^{-\left (\int \frac {2 \alpha \lambda x^{k}-\beta x^{s}}{a x^{n}+b x^{m}+c}d x \right )}}{a x^{n}+b x^{m}+c}d x \right )-{\mathrm e}^{-\left (\int \frac {2 \alpha \lambda x^{k}-\beta x^{s}}{a x^{n}+b x^{m}+c}d x \right )}}{\lambda +y}\right )\]

________________________________________________________________________________________

6.2.5.47 [523] problem number 47

problem number 523

Added January 2, 2019.

Problem 2.2.5.47 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x(a x^n + b x^m + c) w_x - \left ( s x^k y^2 -(a x^n + b x^m+c) y - s \lambda x^{k+2} \right ) w_y = 0 \]

Mathematica


\[\left \{\left \{w(x,y)\to c_1\left (\frac {\tanh ^{-1}\left (\frac {y}{\sqrt {\lambda } x}\right )}{\sqrt {\lambda }}-\int _1^x\frac {s K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right )\right \}\right \}\]

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-\sqrt {\lambda }\, s \left (\int \frac {x^{k}}{a x^{n}+b x^{m}+c}d x \right )+\arctanh \left (\frac {y}{\sqrt {\lambda }\, x}\right )}{\sqrt {\lambda }\, s}\right )\]

________________________________________________________________________________________

6.2.5.48 [524] problem number 48

problem number 524

Added January 2, 2019.

Problem 2.2.5.48 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n + b x^m + c) w_x + \left ( (a x^n+b x^m + c)y^2-a n(n-1)x^{n-2}-b m(m-1) x^{m-2}\right ) w_y = 0 \]

Mathematica


Failed

Maple


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

________________________________________________________________________________________

6.2.5.49 [525] problem number 49

problem number 525

Added January 2, 2019.

Problem 2.2.5.49 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n + b y^n + x) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + y \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.50 [526] problem number 50

problem number 526

Added January 2, 2019.

Problem 2.2.5.50 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n + b y^n + A x^2 + B x y) w_x + \left ( \alpha x^k y^{n-k} + \beta x^m y^{n-m} + A x y + B y^2\right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.51 [527] problem number 51

problem number 527

Added January 2, 2019.

Problem 2.2.5.51 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a y^m + b x^n + s) w_x - \left ( \alpha x^k + b n x^{n-1} y + \beta \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-\left (m +1\right ) \alpha x x^{k}-\left (k +1\right ) \left (a y y^{m}+\left (m +1\right ) b y x^{n}+\left (\beta x +s y \right ) \left (m +1\right )\right )}{\left (m +1\right ) \left (k +1\right )}\right )\]

________________________________________________________________________________________

6.2.5.52 [528] problem number 52

problem number 528

Added January 2, 2019.

Problem 2.2.5.52 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n y^m +x) w_x + \left ( b x^k y^{n+m-k} + y \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.53 [529] problem number 53

problem number 529

Added January 2, 2019.

Problem 2.2.5.53 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x(a x^n y^m +\alpha ) w_x - y \left ( b x^n y^m + \beta \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (x^{\left (a n -b m \right ) \beta m} \left (y^{m}\right )^{\left (a n -b m \right ) \alpha } \left (\alpha n -\beta m +\left (a n -b m \right ) x^{n} y^{m}\right )^{-\left (a \beta -\alpha b \right ) m}\right )\]

________________________________________________________________________________________

6.2.5.54 [530] problem number 54

problem number 530

Added January 2, 2019.

Problem 2.2.5.54 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x(a n x^k y^{n+k} + s) w_x - y \left ( b m x^{m+k} y^k + s \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.55 [531] problem number 55

problem number 531

Added January 2, 2019.

Problem 2.2.5.55 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n y^m + A x^2 + B x y) w_x + \left ( b x^k y^{n+m-k} + A x y+ B y^2 \right ) w_y = 0 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

6.2.5.56 [532] problem number 56

problem number 532

Added January 2, 2019.

Problem 2.2.5.56 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ (a x^n y^m + b x y^k) w_x + \left ( \alpha y^s + \beta \right ) w_y = 0 \]

Mathematica


Failed

Maple


\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int \frac {y^{m} {\mathrm e}^{\left (n -1\right ) b \left (\int \frac {y^{k}}{\alpha y^{s}+\beta }d y \right )}}{\alpha y^{s}+\beta }d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \frac {y^{k}}{\alpha y^{s}+\beta }d y \right )}\right )\]

________________________________________________________________________________________