36 HFOPDE, chapter 2.2.2

 36.1 problem number 1
 36.2 problem number 2
 36.3 problem number 3
 36.4 problem number 4
 36.5 problem number 5
 36.6 problem number 6
 36.7 problem number 7
 36.8 problem number 8
 36.9 problem number 9
 36.10 problem number 10
 36.11 problem number 11
 36.12 problem number 12
 36.13 problem number 13
 36.14 problem number 14
 36.15 problem number 15
 36.16 problem number 16
 36.17 problem number 17
 36.18 problem number 18
 36.19 problem number 19
 36.20 problem number 20
 36.21 problem number 21
 36.22 problem number 22
 36.23 problem number 23
 36.24 problem number 24
 36.25 problem number 25
 36.26 problem number 26
 36.27 problem number 27
 36.28 problem number 28
 36.29 problem number 29
 36.30 problem number 30
 36.31 problem number 31, Hesse’s equation

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36.1 problem number 1

problem number 223

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

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

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36.2 problem number 2

problem number 224

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

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

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36.3 problem number 3

problem number 225

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

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

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36.4 problem number 4

problem number 226

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

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

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36.5 problem number 5

problem number 227

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

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

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36.6 problem number 6

problem number 228

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

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

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36.7 problem number 7

problem number 229

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

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

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36.8 problem number 8

problem number 230

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

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

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36.9 problem number 9

problem number 231

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

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

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36.10 problem number 10

problem number 232

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

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

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36.11 problem number 11

problem number 233

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

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

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36.12 problem number 12

problem number 234

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

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

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36.13 problem number 13

problem number 235

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

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

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36.14 problem number 14

problem number 236

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

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

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36.15 problem number 15

problem number 237

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

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

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36.16 problem number 16

problem number 238

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

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

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36.17 problem number 17

problem number 239

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

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

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36.18 problem number 18

problem number 240

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

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

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36.19 problem number 19

problem number 241

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

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

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36.20 problem number 20

problem number 242

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

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^2+b x^2\right )+2 b x w^{(0,1)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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36.21 problem number 21

problem number 243

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

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

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36.22 problem number 22

problem number 244

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

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

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36.23 problem number 23

problem number 245

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

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

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36.24 problem number 24

problem number 246

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

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a y^2+b x^2+c y\right )+2 b x w^{(0,1)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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36.25 problem number 25

problem number 247

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

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

Maple

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

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36.26 problem number 26

problem number 248

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

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (A k y+A x y+B k x+B x^2\right )+w^{(0,1)}(x,y) \left (k y (\text{D0}-B)+\text{C0} y^2+\text{D0} x y\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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36.27 problem number 27

problem number 249

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

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (A y^2+B x y+\text{C0} x^2+k x\right )+w^{(0,1)}(x,y) \left (\text{D0} y^2+\text{E0} x y+\text{F0} x^2+k y\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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36.28 problem number 28

problem number 250

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

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (A y^2+B x y+\text{C0} x^2\right )+w^{(0,1)}(x,y) \left (\text{D0} y^2+\text{E0} x y+\text{F0} x^2\right )=0,w(x,y),\{x,y\}\right ] \]

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

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36.29 problem number 29

problem number 251

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

\[ \text{DSolve}\left [w^{(1,0)}(x,y) \left (a+A y^2+2 B x y+\text{D0} x^2\right )-w^{(0,1)}(x,y) \left (-b+2 \text{D0} x y+\text{D0} y^2-\text{E0} x^2\right )=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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36.30 problem number 30

problem number 252

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

\[ \text{DSolve}\left [\left (a y+x^2-2 x y+y^2\right ) w^{(1,0)}(x,y)+a y w^{(0,1)}(x,y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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

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36.31 problem number 31, Hesse’s equation

problem number 253

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

\[ \text{DSolve}\left [w^{(1,0)}(x,y) (x (\text{a1}+\text{b1} x+\text{c1} y)-\text{a2}-\text{b2} x-\text{c2} y)+w^{(0,1)}(x,y) (y (\text{a1}+\text{b1} x+\text{c1} y)-\text{a3}-\text{b3} x-\text{c3} y)=0,w(x,y),\{x,y\}\right ] \]

Maple

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