39 HFOPDE, chapter 2.2.2

 39.1 problem number 1
 39.2 problem number 2
 39.3 problem number 3
 39.4 problem number 4
 39.5 problem number 5
 39.6 problem number 6
 39.7 problem number 7
 39.8 problem number 8
 39.9 problem number 9
 39.10 problem number 10
 39.11 problem number 11
 39.12 problem number 12
 39.13 problem number 13
 39.14 problem number 14
 39.15 problem number 15
 39.16 problem number 16
 39.17 problem number 17
 39.18 problem number 18
 39.19 problem number 19
 39.20 problem number 20
 39.21 problem number 21
 39.22 problem number 22
 39.23 problem number 23
 39.24 problem number 24
 39.25 problem number 25
 39.26 problem number 26
 39.27 problem number 27
 39.28 problem number 28
 39.29 problem number 29
 39.30 problem number 30
 39.31 problem number 31, Hesse’s equation

_______________________________________________________________________________________

39.1 problem number 1

problem number 239

Added January 2, 2019.

Problem 2.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{1}{6} \left (-2 a x^3-3 b x^2-6 c x+6 y\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{x}^{3}-1/2\,b{x}^{2}-cx+y \right ) \]

_______________________________________________________________________________________

39.2 problem number 2

problem number 240

Added January 2, 2019.

Problem 2.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{4\,ca-{b}^{2}}} \left ( -x\sqrt{4\,ca-{b}^{2}}+2\,\arctan \left ({\frac{2\,ya+b}{\sqrt{4\,ca-{b}^{2}}}} \right ) \right ) } \right ) \]

_______________________________________________________________________________________

39.3 problem number 3

problem number 241

Added January 2, 2019.

Problem 2.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ \left ( b{x}^{2}{a}^{2}+y{a}^{3}+{a}^{2}cx+2\,axb+ca+2\,b \right ){{\rm e}^{-ax}}}{{a}^{3}}} \right ) \]

_______________________________________________________________________________________

39.4 problem number 4

problem number 242

Added January 2, 2019.

Problem 2.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{a x^2}{2}-k x} \left (2 a^{3/2} b x+2 a^{3/2} c+2 a^{5/2} y-\sqrt{2 \pi } a^2 s e^{\frac{(a x+k)^2}{2 a}} \text{Erf}\left (\frac{k}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{a} x}{\sqrt{2}}\right )-\sqrt{2 \pi } b k^2 e^{\frac{(a x+k)^2}{2 a}} \text{Erf}\left (\frac{k}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{a} x}{\sqrt{2}}\right )-\sqrt{2 \pi } a b e^{\frac{(a x+k)^2}{2 a}} \text{Erf}\left (\frac{k}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{a} x}{\sqrt{2}}\right )-2 \sqrt{a} b k+\sqrt{2 \pi } a c k e^{\frac{(a x+k)^2}{2 a}} \text{Erf}\left (\frac{k}{\sqrt{2} \sqrt{a}}+\frac{\sqrt{a} x}{\sqrt{2}}\right )\right )}{2 a^{5/2}}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{{{\rm e}^{-x \left ( ax+2\,k \right ) }}}{{a}^{5/2}} \left ( \sqrt{2}{{\rm e}^{1/2\,{\frac{2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac{\sqrt{2} \left ( ax+k \right ) }{\sqrt{a}}} \right ){a}^{2}s\sqrt{\pi }-\sqrt{2}{{\rm e}^{1/2\,{\frac{2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac{\sqrt{2} \left ( ax+k \right ) }{\sqrt{a}}} \right ) ack\sqrt{\pi }+\sqrt{2}{{\rm e}^{1/2\,{\frac{2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac{\sqrt{2} \left ( ax+k \right ) }{\sqrt{a}}} \right ) b{k}^{2}\sqrt{\pi }+\sqrt{2}{{\rm e}^{1/2\,{\frac{2\,{a}^{2}{x}^{2}+4\,akx+{k}^{2}}{a}}}}\erf \left ( 1/2\,{\frac{\sqrt{2} \left ( ax+k \right ) }{\sqrt{a}}} \right ) ab\sqrt{\pi }-2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}y{a}^{5/2}-2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}{a}^{3/2}bx-2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}{a}^{3/2}c+2\,{{\rm e}^{1/2\,x \left ( ax+2\,k \right ) }}bk\sqrt{a} \right ) } \right ) \]

_______________________________________________________________________________________

39.5 problem number 5

problem number 243

Added January 2, 2019.

Problem 2.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-y \text{ParabolicCylinderD}\left (-2,i \sqrt{2} \sqrt{a} x\right )+a x \text{ParabolicCylinderD}\left (-2,i \sqrt{2} \sqrt{a} x\right )+i \sqrt{2} \sqrt{a} \text{ParabolicCylinderD}\left (-1,i \sqrt{2} \sqrt{a} x\right )}{y \text{ParabolicCylinderD}\left (1,\sqrt{2} \sqrt{a} x\right )+a x \text{ParabolicCylinderD}\left (1,\sqrt{2} \sqrt{a} x\right )-\sqrt{2} \sqrt{a} \text{ParabolicCylinderD}\left (2,\sqrt{2} \sqrt{a} x\right )}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{-a{x}^{2}+yx+1}{\sqrt{\pi }\erf \left ( \sqrt{-a}x \right ) \left ( -a \right ) ^{3/2}{x}^{2}+\sqrt{\pi }\erf \left ( \sqrt{-a}x \right ) \sqrt{-a}yx-{{\rm e}^{a{x}^{2}}}ax+\sqrt{\pi }\erf \left ( \sqrt{-a}x \right ) \sqrt{-a}+{{\rm e}^{a{x}^{2}}}y}} \right ) \]

_______________________________________________________________________________________

39.6 problem number 6

problem number 244

Added January 2, 2019.

Problem 2.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{-\sqrt{\pi } y \text{Erfi}\left (\sqrt{a} x\right )+\sqrt{\pi } a x \text{Erfi}\left (\sqrt{a} x\right )-2 \sqrt{a} e^{a x^2}}{2 \sqrt{a} (a x-y)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ \left ( -ax+y \right ) \sqrt{\pi }}{\sqrt{\pi }\erf \left ( \sqrt{-a}x \right ) ax-\sqrt{\pi }\erf \left ( \sqrt{-a}x \right ) y-2\,\sqrt{-a}{{\rm e}^{a{x}^{2}}}}} \right ) \]

_______________________________________________________________________________________

39.7 problem number 7

problem number 245

Added January 2, 2019.

Problem 2.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{\sqrt{2 \pi } \sqrt{a} x y \text{Erfi}\left (\frac{\sqrt{a} x}{\sqrt{2}}\right )+\sqrt{2 \pi } \sqrt{a} \text{Erfi}\left (\frac{\sqrt{a} x}{\sqrt{2}}\right )-2 y e^{\frac{a x^2}{2}}}{2 \sqrt{2} \sqrt{a} (x y+1)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{yx+1} \left ( -\erf \left ( 1/2\,\sqrt{-2\,a}x \right ) yax+{{\rm e}^{1/2\,a{x}^{2}}}\sqrt{-2\,{\frac{a}{\pi }}}y-a\erf \left ( 1/2\,\sqrt{-2\,a}x \right ) \right ){\frac{1}{\sqrt{-2\,{\frac{a}{\pi }}}}}} \right ) \]

_______________________________________________________________________________________

39.8 problem number 8

problem number 246

Added January 2, 2019.

Problem 2.2.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{\sqrt{2}}{\sqrt{-a} \left ( b-y \right ) } \left ({{\rm e}^{1/2\,{\frac{ \left ( ax+2\,b \right ) ^{2}}{a}}}}\sqrt{2}\sqrt{-a}+\sqrt{\pi }\erf \left ( 1/2\,{\frac{ \left ( ax+2\,b \right ) \sqrt{2}}{\sqrt{-a}}} \right ) b-\sqrt{\pi }\erf \left ( 1/2\,{\frac{ \left ( ax+2\,b \right ) \sqrt{2}}{\sqrt{-a}}} \right ) y \right ){{\rm e}^{-2\,{\frac{{b}^{2}}{a}}}}} \right ) \]

_______________________________________________________________________________________

39.9 problem number 9

problem number 247

Added January 2, 2019.

Problem 2.2.2.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{xa\sqrt{k}-\arctan \left ( ya\sqrt{k}+xa\sqrt{k}+\sqrt{k}c \right ) }{a\sqrt{k}}} \right ) \]

_______________________________________________________________________________________

39.10 problem number 10

problem number 248

Added January 2, 2019.

Problem 2.2.2.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{ca}} \left ( -x\sqrt{ca}+\arctan \left ({\frac{ya}{x\sqrt{ca}}} \right ) \right ) } \right ) \]

_______________________________________________________________________________________

39.11 problem number 11

problem number 249

Added January 2, 2019.

Problem 2.2.2.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{4\,ca-{b}^{2}}} \left ( -x\sqrt{4\,ca-{b}^{2}}+2\,\arctan \left ({\frac{2\,ya+bx}{x\sqrt{4\,ca-{b}^{2}}}} \right ) \right ) } \right ) \]

_______________________________________________________________________________________

39.12 problem number 12

problem number 250

Added January 2, 2019.

Problem 2.2.2.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x + c) w_x + \left ( \alpha (a y+b x)^2+\beta ( a y+b x) - b x+\gamma \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{1}{2} \left (2 \tan ^{-1}\left (\frac{2 a^3 \alpha ^2 y \sqrt{\frac{4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}+2 a^2 \alpha ^2 b x \sqrt{\frac{4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}+a^2 \alpha \beta \sqrt{\frac{4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}}{4 a \alpha \gamma -a \beta ^2+4 \alpha b c}\right )-a \alpha \log (a x+c) \sqrt{\frac{4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{{a}^{3} \left ( 4\,ga\alpha -a{\beta }^{2}+4\,\alpha \,bc \right ) }} \left ( -2\,\arctan \left ({\frac{{a}^{2} \left ( 2\,ya\alpha +2\,\alpha \,bx+\beta \right ) }{\sqrt{4\,{a}^{4}\alpha \,g-{a}^{4}{\beta }^{2}+4\,{a}^{3}\alpha \,bc}}} \right ){a}^{2}+\ln \left ( ax+c \right ) \sqrt{{a}^{3} \left ( 4\,ga\alpha -a{\beta }^{2}+4\,\alpha \,bc \right ) } \right ) } \right ) \]

_______________________________________________________________________________________

39.13 problem number 13

problem number 251

Added January 2, 2019.

Problem 2.2.2.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{-ax+by}{axy}} \right ) \]

_______________________________________________________________________________________

39.14 problem number 14

problem number 252

Added January 2, 2019.

Problem 2.2.2.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^2+b) w_x - \left ( y^2-2 x y+(1-a)x^2 -b \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\sqrt{ab} \left ( x-y \right ) } \left ( \arctan \left ({\frac{ax}{\sqrt{ab}}} \right ) y-\arctan \left ({\frac{ax}{\sqrt{ab}}} \right ) x-\sqrt{ab} \right ) } \right ) \]

_______________________________________________________________________________________

39.15 problem number 15

problem number 253

Added January 2, 2019.

Problem 2.2.2.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a_1 x^2+b_1 x + x_1) w_x + (a_2 y^2+b_2 y+c_2) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{2 \left (\frac{4 \text{a2} \text{c2} \sqrt{4 \text{a1} \text{c1}-\text{b1}^2} \tan ^{-1}\left (\frac{2 \text{a1} x+\text{b1}}{\sqrt{4 \text{a1} \text{c1}-\text{b1}^2}}\right )}{\text{b1}^2-4 \text{a1} \text{c1}}-\frac{\text{b2}^2 \sqrt{4 \text{a1} \text{c1}-\text{b1}^2} \tan ^{-1}\left (\frac{2 \text{a1} x+\text{b1}}{\sqrt{4 \text{a1} \text{c1}-\text{b1}^2}}\right )}{\text{b1}^2-4 \text{a1} \text{c1}}+\sqrt{4 \text{a2} \text{c2}-\text{b2}^2} \tan ^{-1}\left (\frac{2 \text{a2} y \sqrt{4 \text{a2} \text{c2}-\text{b2}^2}+\text{b2} \sqrt{4 \text{a2} \text{c2}-\text{b2}^2}}{4 \text{a2} \text{c2}-\text{b2}^2}\right )\right )}{4 \text{a2} \text{c2}-\text{b2}^2}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -2\,{\frac{1}{\sqrt{4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}}\sqrt{4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}} \left ( \sqrt{4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}\arctan \left ({\frac{2\,{\it a1}\,x+{\it b1}}{\sqrt{4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}}}} \right ) -\arctan \left ({\frac{2\,{\it a2}\,y+{\it b2}}{\sqrt{4\,{\it c2}\,{\it a2}-{{\it b2}}^{2}}}} \right ) \sqrt{4\,{\it c1}\,{\it a1}-{{\it b1}}^{2}} \right ) } \right ) \]

_______________________________________________________________________________________

39.16 problem number 16

problem number 254

Added January 2, 2019.

Problem 2.2.2.16 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{ \left ( b-x \right ) ^{k} \left ( -kb+kx+ky+y \right ) \left ( a-x \right ) ^{-k}}{-ak+kx+ky+y}} \right ) \]

_______________________________________________________________________________________

39.17 problem number 17

problem number 255

Added January 2, 2019.

Problem 2.2.2.17 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a_1 y^2+b_1 y + c_1) w_x +( a_2 x^2+b_2 x+c_2) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{1}{6} \left (2 \text{a1} y^3-2 \text{a2} x^3+3 \text{b1} y^2-3 \text{b2} x^2+6 \text{c1} y-6 \text{c2} x\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( 1/3\,{\it a1}\,{y}^{3}-1/3\,{\it a2}\,{x}^{3}+1/2\,{\it b1}\,{y}^{2}-1/2\,{x}^{2}{\it b2}+{\it c1}\,y-{\it c2}\,x \right ) \]

_______________________________________________________________________________________

39.18 problem number 18

problem number 256

Added January 2, 2019.

Problem 2.2.2.18 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{a}^{2}{y}^{2}-2\,acx-bc}{{a}^{2} \left ({a}^{2}{x}^{2}+2\,axb+{b}^{2} \right ) }} \right ) \]

_______________________________________________________________________________________

39.19 problem number 19

problem number 257

Added January 2, 2019.

Problem 2.2.2.19 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{1}{12} \left (4 a y^3+12 b x y+3 x^4\right )\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{y}^{3}-1/4\,{x}^{4}-bxy \right ) \]

_______________________________________________________________________________________

39.20 problem number 20

problem number 258

Added January 2, 2019.

Problem 2.2.2.20 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Failed} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ \left ({y}^{2}a+b{x}^{2}+2\,ya+2\,a \right ){{\rm e}^{-y}}}{b}} \right ) \]

_______________________________________________________________________________________

39.21 problem number 21

problem number 259

Added January 2, 2019.

Problem 2.2.2.21 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

39.22 problem number 22

problem number 260

Added January 2, 2019.

Problem 2.2.2.22 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,a{y}^{3}+1/3\,b{x}^{3}-y{x}^{2}+cx \right ) \]

_______________________________________________________________________________________

39.23 problem number 23

problem number 261

Added January 2, 2019.

Problem 2.2.2.23 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{A{y}^{2}+{a}^{2}B-B{x}^{2}-Cyx}{y}} \right ) \]

_______________________________________________________________________________________

39.24 problem number 24

problem number 262

Added January 2, 2019.

Problem 2.2.2.24 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Failed} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ \left ({y}^{2}a+b{x}^{2}+2\,ya+cy+2\,a+c \right ){{\rm e}^{-y}}}{b}} \right ) \]

_______________________________________________________________________________________

39.25 problem number 25

problem number 263

Added January 2, 2019.

Problem 2.2.2.25 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A x y+B x^2+k x) w_x +(D y^2+E x y+F x^2+k y)w_y = 0 \]

Mathematica

\[ \text{\$Aborted} \] Timed out

Maple

\[ \text{ Exception } \] Timed out

_______________________________________________________________________________________

39.26 problem number 26

problem number 264

Added January 2, 2019.

Problem 2.2.2.26 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A x y+A k y+B x^2+B k x) w_x +(C y^2+D x y+k(D-B)y) w_y = 0 \]

Mathematica

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

_______________________________________________________________________________________

39.27 problem number 27

problem number 265

Added January 2, 2019.

Problem 2.2.2.27 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A y^2+B x y+C x^2+k x) w_x +(D y^2+E x y + F x^2+k y) w_y = 0 \]

Mathematica

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

_______________________________________________________________________________________

39.28 problem number 28

problem number 266

Added January 2, 2019.

Problem 2.2.2.28 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (A y^2+B x y+C x^2) w_x +(D y^2+E x y + F x^2) w_y = 0 \]

Mathematica

\[ \text{Failed} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\sum _{{\it \_R}=\RootOf \left ( A{{\it \_Z}}^{3}+ \left ( B-{\it D0} \right ){{\it \_Z}}^{2}+ \left ({\it C0}-{\it E0} \right ){\it \_Z}-{\it F0} \right ) }{\frac{A{{\it \_R}}^{2}+B{\it \_R}+{\it C0}}{3\,A{{\it \_R}}^{2}+2\,B{\it \_R}-2\,{\it D0}\,{\it \_R}+{\it C0}-{\it E0}}\ln \left ({\frac{-{\it \_R}\,x+y}{x}} \right ) }-\ln \left ( x \right ) \right ) \] solution contains RootOf

_______________________________________________________________________________________

39.29 problem number 29

problem number 267

Added January 2, 2019.

Problem 2.2.2.29 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Failed} \]

Maple

\[ \text{ sol=() } \]

_______________________________________________________________________________________

39.30 problem number 30

problem number 268

Added January 2, 2019.

Problem 2.2.2.30 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{Failed} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{\ln \left ( y \right ) x-\ln \left ( y \right ) y+a}{x-y}} \right ) \]

_______________________________________________________________________________________

39.31 problem number 31, Hesse’s equation

problem number 269

Added January 2, 2019.

Problem 2.2.2.31 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux. Reference E. Kamke (1965).

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

\[ (x f_1-f_2) w_x +(y f_1-f_3)w_y = 0 \] Where \(f_n = a_n+b_n x + c_n y\).

Mathematica

\[ \text{Failed} \]

Maple

\[ \text{Too large to display} \]