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

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39.1 problem number 1

problem number 239

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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39.2 problem number 2

problem number 240

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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39.3 problem number 3

problem number 241

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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39.4 problem number 4

problem number 242

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

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

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39.5 problem number 5

problem number 243

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

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

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39.6 problem number 6

problem number 244

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

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

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39.7 problem number 7

problem number 245

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

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

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39.8 problem number 8

problem number 246

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

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

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39.9 problem number 9

problem number 247

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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39.10 problem number 10

problem number 248

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

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

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39.11 problem number 11

problem number 249

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

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

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39.12 problem number 12

problem number 250

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

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

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39.13 problem number 13

problem number 251

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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39.14 problem number 14

problem number 252

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

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

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39.15 problem number 15

problem number 253

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

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

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39.16 problem number 16

problem number 254

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

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

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39.17 problem number 17

problem number 255

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

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

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39.18 problem number 18

problem number 256

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

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

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39.19 problem number 19

problem number 257

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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39.20 problem number 20

problem number 258

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

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

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39.21 problem number 21

problem number 259

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

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

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39.22 problem number 22

problem number 260

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

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

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39.23 problem number 23

problem number 261

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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39.24 problem number 24

problem number 262

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

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

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39.25 problem number 25

problem number 263

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

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39.26 problem number 26

problem number 264

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

$\text{ sol=() }$

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39.27 problem number 27

problem number 265

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

$\text{ sol=() }$

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39.28 problem number 28

problem number 266

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

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39.29 problem number 29

problem number 267

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

$\text{ sol=() }$

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39.30 problem number 30

problem number 268

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

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

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39.31 problem number 31, Hesse’s equation

problem number 269

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

$\text{Too large to display}$