### 41 HFOPDE, chapter 2.3.2

41.1 problem number 1
41.2 problem number 2
41.3 problem number 3
41.4 problem number 4
41.5 problem number 5
41.6 problem number 6
41.7 problem number 7
41.8 problem number 8
41.9 problem number 9
41.10 problem number 10
41.11 problem number 11
41.12 problem number 12
41.13 problem number 13
41.14 problem number 14
41.15 problem number 15
41.16 problem number 16
41.17 problem number 17
41.18 problem number 18
41.19 problem number 19
41.20 problem number 20
41.21 problem number 21
41.22 problem number 22
41.23 problem number 23
41.24 problem number 24
41.25 problem number 25
41.26 problem number 26
41.27 problem number 27
41.28 problem number 28
41.29 problem number 29
41.30 problem number 30
41.31 problem number 31
41.32 problem number 32
41.33 problem number 33
41.34 problem number 34
41.35 problem number 35
41.36 problem number 36

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#### 41.1 problem number 1

problem number 337

Problem 2.3.2.1 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( y^2+a \lambda e^{\lambda x}- a^2 e^{2 \lambda x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{y \text{ExpIntegralEi}\left (\frac{2 a e^{\lambda x}}{\lambda }\right )-a e^{\lambda x} \text{ExpIntegralEi}\left (\frac{2 a e^{\lambda x}}{\lambda }\right )+\lambda e^{\frac{2 a e^{\lambda x}}{\lambda }}}{a e^{\lambda x}-y}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{(-a{{\rm e}^{\lambda \,x}}+y) \left ( -{{\rm e}^{\lambda \,x}}\Ei \left ( 1,-2\,{\frac{a{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) a-{{\rm e}^{2\,{\frac{a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\lambda +y\Ei \left ( 1,-2\,{\frac{a{{\rm e}^{\lambda \,x}}}{\lambda }} \right ) \right ) ^{-1}} \right )$

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#### 41.2 problem number 2

problem number 338

Problem 2.3.2.2 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( y^2+b y+ a (\lambda -b) e^{\lambda x} - a^2 e^{2 \lambda x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (-\frac{2^{b/\lambda } \lambda ^{-\frac{b}{\lambda }} e^{b x} a^{b/\lambda } \left (-y \text{LaguerreL}\left (-\frac{b}{\lambda },\frac{b}{\lambda },\frac{2 a e^{\lambda x}}{\lambda }\right )+2 a e^{\lambda x} \text{LaguerreL}\left (-\frac{b}{\lambda }-1,\frac{b}{\lambda }+1,\frac{2 a e^{\lambda x}}{\lambda }\right )+a e^{\lambda x} \text{LaguerreL}\left (-\frac{b}{\lambda },\frac{b}{\lambda },\frac{2 a e^{\lambda x}}{\lambda }\right )-b \text{LaguerreL}\left (-\frac{b}{\lambda },\frac{b}{\lambda },\frac{2 a e^{\lambda x}}{\lambda }\right )\right )}{a e^{\lambda x}-y}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{-a{{\rm e}^{\lambda \,x}}+y} \left ( -{{\rm e}^{\lambda \,x}}a\int \!{{\rm e}^{{\frac{bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x+y\int \!{{\rm e}^{{\frac{bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x+{{\rm e}^{{\frac{bx\lambda +2\,a{{\rm e}^{\lambda \,x}}}{\lambda }}}} \right ) } \right )$

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#### 41.3 problem number 3

problem number 339

Problem 2.3.2.3 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x + \left ( y^2+a e^{\lambda x} y-a b e^{\lambda x} - b^2 \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{2 b (-1)^{-\frac{b}{\lambda }} \lambda ^{-\frac{2 b}{\lambda }-1} \left (y \lambda ^{\frac{2 b}{\lambda }} \text{Gamma}\left (\frac{2 b}{\lambda },0,-\frac{a e^{\lambda x}}{\lambda }\right )-b \lambda ^{\frac{2 b}{\lambda }} \text{Gamma}\left (\frac{2 b}{\lambda },0,-\frac{a e^{\lambda x}}{\lambda }\right )+\lambda (-1)^{\frac{2 b}{\lambda }} a^{\frac{2 b}{\lambda }} e^{\frac{a e^{\lambda x}}{\lambda }+2 b x}\right )}{b-y}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{-b+y} \left ({{\rm e}^{{\frac{2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}}+y\int \!{{\rm e}^{{\frac{2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x-b\int \!{{\rm e}^{{\frac{2\,bx\lambda +a{{\rm e}^{\lambda \,x}}}{\lambda }}}}\,{\rm d}x \right ) } \right )$

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#### 41.4 problem number 4

problem number 340

Problem 2.3.2.4 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x - \left ( y^2-a x e^{\lambda x} y + a e^{\lambda x} \right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{x{\lambda }^{2} \left ( yx-1 \right ) } \left ( y{x}^{2}\int \!{\frac{1}{{x}^{2}}{{\rm e}^{{\frac{{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}}\,{\rm d}x-{{\rm e}^{{\frac{{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}-x\int \!{\frac{1}{{x}^{2}}{{\rm e}^{{\frac{{{\rm e}^{\lambda \,x}}a \left ( \lambda \,x-1 \right ) }{{\lambda }^{2}}}}}}\,{\rm d}x \right ) } \right )$

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#### 41.5 problem number 5

problem number 341

Problem 2.3.2.5 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left (a e^{\lambda x} y^2 + b e^{-\lambda x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{e^{x \left (-\sqrt{\lambda ^2-4 a b}\right )} \left (\sqrt{\lambda ^2-4 a b}-2 a y e^{\lambda x}-\lambda \right )}{2 a y e^{\lambda x} \sqrt{\lambda ^2-4 a b}+\lambda \sqrt{\lambda ^2-4 a b}-4 a b+\lambda ^2}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{\lambda }{\sqrt{{\lambda }^{2} \left ( 4\,ab-{\lambda }^{2} \right ) }} \left ( 2\,\lambda \,\arctan \left ({\frac{\lambda \, \left ( 2\,{{\rm e}^{\lambda \,x}}ay+\lambda \right ) }{\sqrt{4\,{\lambda }^{2}ab-{\lambda }^{4}}}} \right ) -\sqrt{{\lambda }^{2} \left ( 4\,ab-{\lambda }^{2} \right ) }x \right ) } \right )$

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#### 41.6 problem number 6

problem number 342

Problem 2.3.2.6 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left (a e^{\lambda x} y^2 + b \mu e^{\mu x} - a b^2 e^{(\lambda + 2 \mu )x} \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 41.7 problem number 7

problem number 343

Problem 2.3.2.7 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left (a e^{\lambda x} y^2 + b y + c e^{-\lambda x} \right ) w_y = 0$

Mathematica

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

Maple

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

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#### 41.8 problem number 8

problem number 344

Problem 2.3.2.8 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left (a e^{\lambda x} y^2 + \mu y - a b^2 e^{(\lambda +2 \mu )x} \right ) w_y = 0$

Mathematica

$\text{DSolve}\left [w^{(0,1)}(x,y) \left (-a b^2 e^{x (\lambda +2 \mu )}+a y^2 e^{\lambda x}+\mu y\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ]$ kernel error generated

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( y\sinh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{\lambda \,x}}+{{\rm e}^{x \left ( \lambda +\mu \right ) }}\cosh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) b \right ) \left ( y\cosh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{\lambda \,x}}+{{\rm e}^{x \left ( \lambda +\mu \right ) }}\sinh \left ({\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) b \right ) ^{-1}} \right )$

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#### 41.9 problem number 9

problem number 345

Problem 2.3.2.9 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left (e^{\lambda x} y^2 + a e^{\mu x} y+a \lambda e^{(\mu -lambda)x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{\mu ^{-\frac{\lambda }{\mu }} e^{-\lambda x} \left (y (-1)^{\lambda /\mu } e^{2 \lambda x} a^{\lambda /\mu } \text{Gamma}\left (-\frac{\lambda }{\mu },-\frac{a e^{\mu x}}{\mu }\right )+\lambda (-1)^{\lambda /\mu } e^{\lambda x} a^{\lambda /\mu } \text{Gamma}\left (-\frac{\lambda }{\mu },-\frac{a e^{\mu x}}{\mu }\right )+\mu ^{\frac{\lambda }{\mu }+1} \left (-e^{\frac{a e^{\mu x}}{\mu }}\right )\right )}{y e^{\lambda x}+\lambda }\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{{{\rm e}^{\lambda \,x}} \left ( y{{\rm e}^{\lambda \,x}}\lambda -y{{\rm e}^{\lambda \,x}}\mu +{\lambda }^{2}-\lambda \,\mu \right ) \left ({{\rm e}^{\mu \,x}}{\mbox{_1F_1}(-{\frac{\lambda -\mu }{\mu }};\,-{\frac{\lambda -2\,\mu }{\mu }};\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }})}a\lambda +y{{\rm e}^{\lambda \,x}}{\mbox{_1F_1}(-{\frac{\lambda }{\mu }};\,-{\frac{\lambda -\mu }{\mu }};\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }})}\lambda -y{{\rm e}^{\lambda \,x}}{\mbox{_1F_1}(-{\frac{\lambda }{\mu }};\,-{\frac{\lambda -\mu }{\mu }};\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }})}\mu \right ) ^{-1}} \right )$

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#### 41.10 problem number 10

problem number 346

Problem 2.3.2.10 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x -\left ( \lambda e^{\lambda x} y^2 - a e^{\mu x} y+a e^{(\mu -lambda)x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{\mu \left (\lambda y e^{\lambda x} \text{LaguerreL}\left (\frac{\lambda ^2}{\mu }-\frac{\lambda }{\mu },\frac{\lambda }{\mu },\frac{a e^{\mu x}}{\mu }\right )+a e^{\mu x} \text{LaguerreL}\left (\frac{\lambda ^2}{\mu }-\frac{\lambda }{\mu }-1,\frac{\lambda }{\mu }+1,\frac{a e^{\mu x}}{\mu }\right )-\lambda \text{LaguerreL}\left (\frac{\lambda ^2}{\mu }-\frac{\lambda }{\mu },\frac{\lambda }{\mu },\frac{a e^{\mu x}}{\mu }\right )\right )}{\lambda \left (-\mu y e^{\lambda x} \text{HypergeometricU}\left (\frac{\lambda }{\mu }-\frac{\lambda ^2}{\mu },\frac{\lambda }{\mu }+1,\frac{a e^{\mu x}}{\mu }\right )+\mu \text{HypergeometricU}\left (\frac{\lambda }{\mu }-\frac{\lambda ^2}{\mu },\frac{\lambda }{\mu }+1,\frac{a e^{\mu x}}{\mu }\right )-a e^{\mu x} \text{HypergeometricU}\left (-\frac{\lambda ^2}{\mu }+\frac{\lambda }{\mu }+1,\frac{\lambda }{\mu }+2,\frac{a e^{\mu x}}{\mu }\right )+a \lambda e^{\mu x} \text{HypergeometricU}\left (-\frac{\lambda ^2}{\mu }+\frac{\lambda }{\mu }+1,\frac{\lambda }{\mu }+2,\frac{a e^{\mu x}}{\mu }\right )\right )}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{1 \left ( -y{{\rm e}^{\lambda \,x}}{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda +{{\rm e}^{\mu \,x}}{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}a-{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}+{{\sl M}\left (-{\frac{{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}+{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda -{{\sl M}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu +{{\sl M}\left (-{\frac{{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu \right ) \left ( -y{{\rm e}^{\lambda \,x}}{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda +{{\rm e}^{\mu \,x}}{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}a-{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}{\lambda }^{2}+{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\lambda -{{\sl U}\left (-{\frac{\lambda \, \left ( \lambda -1 \right ) }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu -{{\sl U}\left (-{\frac{{\lambda }^{2}-\lambda +\mu }{\mu }},\,{\frac{\lambda +\mu }{\mu }},\,{\frac{a{{\rm e}^{\mu \,x}}}{\mu }}\right )}\mu \right ) ^{-1}} \right )$

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#### 41.11 problem number 11

problem number 347

Problem 2.3.2.11 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{\lambda x} y^2+ a b e^{(\lambda + \mu )x} y - b \mu e^{\mu x}\right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{a \left ({{\rm e}^{\lambda \,x}}y+{{\rm e}^{x \left ( \lambda +\mu \right ) }}b \right ){{\rm e}^{{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }}}} \left ( -4\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{3}- \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\mu }^{3}+12\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{3}+2\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\mu }^{3}-2\,{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){\lambda }^{3}-{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){\mu }^{3}+y \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\mu }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\,y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\lambda }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\mu }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\mu }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+{a}^{2}{b}^{2} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \lambda \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+{a}^{2}{b}^{2} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+4\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\lambda }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+ \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\mu }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\, \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab{\lambda }^{2}{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+2\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab{\lambda }^{2}+ \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab{\mu }^{2}-5\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}\lambda \,{\mu }^{2}+20\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{2}\mu +11\, \WhittakerM \left ( 1/2\,{\frac{4\,\lambda +3\,\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}\lambda \,{\mu }^{2}-5\,{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){\lambda }^{2}\mu -4\,{{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}} \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) \lambda \,{\mu }^{2}-8\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}{\lambda }^{2}\mu +4\,y \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a{\lambda }^{2}{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+4\,y \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a\lambda \,\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+3\,y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) a\lambda \,\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+4\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab\lambda \,\mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+3\, \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ) ab\lambda \,\mu \,{{\rm e}^{x \left ( \lambda +\mu \right ) +1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-2\,x{\lambda }^{2}-5\,\mu \,x\lambda -3\,{\mu }^{2}x}{\lambda +\mu }}}}+y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){a}^{2}b\lambda \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+y \WhittakerM \left ( -1/2\,{\frac{\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){a}^{2}b\mu \,{{\rm e}^{\lambda \,x+1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}+3\, \WhittakerM \left ( 1/2\,{\frac{2\,\lambda +\mu }{\lambda +\mu }},1/2\,{\frac{3\,\lambda +2\,\mu }{\lambda +\mu }},{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}}{\lambda +\mu }} \right ){{\rm e}^{1/2\,{\frac{ab{{\rm e}^{x \left ( \lambda +\mu \right ) }}-\mu \,x\lambda -{\mu }^{2}x}{\lambda +\mu }}}}ab\lambda \,\mu \right ) ^{-1}} \right )$

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#### 41.12 problem number 12

problem number 348

Problem 2.3.2.12 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{(2 \lambda + \mu ) x} y^2+ \left (b e^{(\lambda + \mu )x} -\lambda \right ) y + c e^{\mu x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{i \pi e^{-\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}} \left (\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}-2 a y e^{x (2 \lambda +\mu )}+b \left (-e^{x (\lambda +\mu )}\right )\right )}{2 \left (2 a y e^{x (2 \lambda +\mu )} \cosh \left (\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )+\sqrt{b^2-4 a c} e^{x (\lambda +\mu )} \sinh \left (\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )+b e^{x (\lambda +\mu )} \cosh \left (\frac{\sqrt{b^2-4 a c} e^{x (\lambda +\mu )}}{2 (\lambda +\mu )}\right )\right )}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \left ( \lambda +\mu \right ) } \left ( 2\,b\lambda \,\arctan \left ({\frac{b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) +2\,b\mu \,\arctan \left ({\frac{b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) -{{\rm e}^{x \left ( \lambda +\mu \right ) }}\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) } \right ) } \right )$

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#### 41.13 problem number 13

problem number 349

Problem 2.3.2.13 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( e^{\lambda x} \left ( y- b e^{\mu x} \right )^2 + b \mu e^{\mu x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{b e^{\lambda x+\mu x}-y e^{\lambda x}-\lambda }{\lambda \left (b e^{\mu x}-y\right )}\right )\right \}\right \}$

Maple

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

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#### 41.14 problem number 14

problem number 350

Problem 2.3.2.14 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{\lambda x} y^2+ b n x^{n-1} - a b^2 e^{\lambda x} x^{2 n} \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 41.15 problem number 15

problem number 351

Problem 2.3.2.15 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( e^{\lambda x} y^2+ a x^n y + a \lambda x^n e^{-\lambda x} \right ) w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{{{\rm e}^{\lambda \,x}}y+\lambda } \left ( y{{\rm e}^{\lambda \,x}}\int \!{{\rm e}^{{\frac{x \left ({x}^{n}a-\lambda \,n-\lambda \right ) }{n+1}}}}\,{\rm d}x+\int \!{{\rm e}^{{\frac{x \left ({x}^{n}a-\lambda \,n-\lambda \right ) }{n+1}}}}\,{\rm d}x\lambda +{{\rm e}^{{\frac{x \left ({x}^{n}a-\lambda \,n-\lambda \right ) }{n+1}}}} \right ) } \right )$

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#### 41.16 problem number 16

problem number 352

Problem 2.3.2.16 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( \lambda e^{\lambda x} y^2+ a x^n e^{\lambda x} y - a x^n e^{2 \lambda x} \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 41.17 problem number 17

problem number 353

Problem 2.3.2.17 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{\lambda x} y^2- a b x^n e^{\lambda x} y + b n x^{n-1} \right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -{a \left ( -b{x}^{n}+y \right ) \left ( -{x}^{n}\int \!\lambda \,{{\rm e}^{-{\frac{ab \left ( -\lambda \right ) ^{-n} \left ({x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}+\lambda \,x}}\,{\rm d}xab+\int \!\lambda \,{{\rm e}^{-{\frac{ab \left ( -\lambda \right ) ^{-n} \left ({x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}+\lambda \,x}}\,{\rm d}xya+\lambda \,{{\rm e}^{-{\frac{ab \left ( -\lambda \right ) ^{-n} \left ({x}^{n} \left ( -\lambda \right ) ^{n}n\Gamma \left ( n \right ) \left ( -\lambda \,x \right ) ^{-n}-{x}^{n} \left ( -\lambda \right ) ^{n}{{\rm e}^{\lambda \,x}}-{x}^{n} \left ( -\lambda \right ) ^{n}n \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \right ) }{\lambda }}}} \right ) ^{-1}} \right )$

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#### 41.18 problem number 18

problem number 354

Problem 2.3.2.18 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a x^n y^2 + b \lambda e^{\lambda x} - a b^2 x^n e^{2 \lambda x} \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 41.19 problem number 19

problem number 355

Problem 2.3.2.19 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a x^n y^2 + \lambda y - a b^2 x^n e^{2 \lambda x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (-i \left (a b (-1)^{-n} \lambda ^{-n-1} \text{Gamma}(n+1,-\lambda x)+\tanh ^{-1}\left (\frac{y e^{-\lambda x}}{b}\right )\right )\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-i}{\lambda } \left ( b{x}^{n}\Gamma \left ( n,-\lambda \,x \right ) n \left ( -\lambda \,x \right ) ^{-n}a-bn{x}^{n} \left ( -\lambda \,x \right ) ^{-n}a\Gamma \left ( n \right ) +ab{x}^{n}{{\rm e}^{\lambda \,x}}+\arctanh \left ({\frac{{{\rm e}^{-\lambda \,x}}y}{b}} \right ) \lambda \right ) } \right )$

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#### 41.20 problem number 20

problem number 356

Problem 2.3.2.20 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a x^n y^2 - a b x^n e^{\lambda x} y + b \lambda e^{\lambda x} \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 41.21 problem number 21

problem number 357

Problem 2.3.2.21 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a x^n y^2 - a x^n \left (b e^{\lambda x} + c \right )y + b \lambda e^{\lambda x} \right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 41.22 problem number 22

problem number 358

Problem 2.3.2.22 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a x^n e^{2 \lambda x} y^2 + \left ( b x^n e^{\lambda x} - \lambda \right ) y + c x^n \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{-4 a^{3/2} c^{3/2} (-1)^{-n} \lambda ^{-n-1} \text{Gamma}(n+1,-\lambda x)+\sqrt{a} b^2 \sqrt{c} (-1)^{-n} \lambda ^{-n-1} \text{Gamma}(n+1,-\lambda x)+2 \sqrt{a} \sqrt{c} \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{2 a y e^{\lambda x} \sqrt{4 a c-b^2}-b \sqrt{4 a c-b^2}}{4 a c-b^2}\right )}{4 a c-b^2}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }\lambda } \left ( - \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}n+ \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n \right ) \sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}n-{{\rm e}^{\lambda \,x}}\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}+2\,b\lambda \,\arctan \left ({\frac{b \left ( 2\,{{\rm e}^{\lambda \,x}}ay+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) \right ) } \right )$

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#### 41.23 problem number 23

problem number 359

Problem 2.3.2.23 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{\lambda x} (y- b x^n - c)^2 +b n x^{n-1} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{a b e^{\lambda x} x^n+a c e^{\lambda x}-a y e^{\lambda x}-\lambda }{\lambda \left (b x^n+c-y\right )}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-ab{x}^{n}{{\rm e}^{\lambda \,x}}+{{\rm e}^{\lambda \,x}}ay-{{\rm e}^{\lambda \,x}}ac+\lambda }{\lambda \, \left ( y-b{x}^{n}-c \right ) }} \right )$

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#### 41.24 problem number 24

problem number 360

Problem 2.3.2.24 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( y^2+2 a \lambda x e^{\lambda x^2} - a^2 e^{2\lambda x^2}\right ) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 41.25 problem number 25

problem number 361

Problem 2.3.2.25 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{-\lambda x^2} y^2 + \lambda x y + a b^2 \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{1}{2} \left (2 \tan ^{-1}\left (\frac{y e^{-\frac{\lambda x^2}{2}}}{b}\right )-\frac{\sqrt{2 \pi } a b \text{Erf}\left (\frac{\sqrt{\lambda } x}{\sqrt{2}}\right )}{\sqrt{\lambda }}\right )\right )\right \}\right \}$

Maple

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

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#### 41.26 problem number 26

problem number 362

Problem 2.3.2.26 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a x^n y^2 + \lambda x y + a b^2 x^n e^{\lambda x^2}\right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (-\frac{1}{2} i \left (a b i^{-n} 2^{\frac{n}{2}+\frac{1}{2}} \lambda ^{-\frac{n}{2}-\frac{1}{2}} \text{Gamma}\left (\frac{n}{2}+\frac{1}{2},-\frac{\lambda x^2}{2}\right )+2 i \tan ^{-1}\left (\frac{y e^{-\frac{\lambda x^2}{2}}}{b}\right )\right )\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({2}^{n/2-1/2}{x}^{n+1}ab \left ( -{x}^{2}\lambda \right ) ^{-1/2-n/2}\Gamma \left ( n/2+1/2 \right ) -{2}^{n/2-1/2}{x}^{n+1}ab \left ( -{x}^{2}\lambda \right ) ^{-1/2-n/2}\Gamma \left ( n/2+1/2,-1/2\,{x}^{2}\lambda \right ) -\arctan \left ({\frac{{{\rm e}^{-1/2\,{x}^{2}\lambda }}y}{b}} \right ) \right )$

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#### 41.27 problem number 27

problem number 363

Problem 2.3.2.27 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{2 \lambda x} y^3 + b e^{\lambda x} y^2 + c y+ d e^{-\lambda x}\right ) w_y = 0$

Mathematica

$\text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^3 e^{2 \lambda x}+b y^2 e^{\lambda x}+c y+d e^{-\lambda x}\right )+w^{(1,0)}(x,y)=0,w(x,y),\{x,y\}\right ]$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( x-\sum _{{\it \_R}=\RootOf \left ( a{{\it \_Z}}^{3}+b{{\it \_Z}}^{2}+ \left ( c+\lambda \right ){\it \_Z}+d \right ) }{\frac{\ln \left ({{\rm e}^{\lambda \,x}}y-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}a+2\,{\it \_R}\,b+c+\lambda }} \right )$ Solution contains RootOf

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#### 41.28 problem number 28

problem number 364

Problem 2.3.2.28 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$w_x +\left ( a e^{\lambda x} y^3 + 3 a b e^{\lambda x} y^2 + c y- 2 a b^3 e^{\lambda x} + b c\right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{e^{-\frac{6 a b^2 e^{\lambda x}}{\lambda }} \left (2 y^2 e^{\frac{6 a b^2 e^{\lambda x}}{\lambda }} \int _1^x a \exp \left (-\frac{6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]+4 b y e^{\frac{6 a b^2 e^{\lambda x}}{\lambda }} \left (\int _1^x a \exp \left (-\frac{6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]\right )+2 b^2 e^{\frac{6 a b^2 e^{\lambda x}}{\lambda }} \left (\int _1^x a \exp \left (-\frac{6 a b^2 e^{\lambda K[1]}}{\lambda }+2 c K[1]+\lambda K[1]\right ) \, dK[1]\right )+e^{2 c x}\right )}{(b+y)^2}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{ \left ( b+y \right ) ^{2}} \left ( 2\,{b}^{2}a\int \!{{\rm e}^{-{\frac{6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-x{\lambda }^{2}}{\lambda }}}}\,{\rm d}x+4\,yab\int \!{{\rm e}^{-{\frac{6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-x{\lambda }^{2}}{\lambda }}}}\,{\rm d}x+2\,{y}^{2}a\int \!{{\rm e}^{-{\frac{6\,a{{\rm e}^{\lambda \,x}}{b}^{2}-2\,c\lambda \,x-x{\lambda }^{2}}{\lambda }}}}\,{\rm d}x+{{\rm e}^{2\,cx-6\,{\frac{a{{\rm e}^{\lambda \,x}}{b}^{2}}{\lambda }}}} \right ) } \right )$

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#### 41.29 problem number 29

problem number 365

Problem 2.3.2.29 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x +\left ( a e^{\lambda x} y^2 + k y + a b^2 x^{2 k} e^{\lambda x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (a \sqrt{b^2} x^k (-\lambda x)^{-k} \text{Gamma}(k,-\lambda x)+\tan ^{-1}\left (\frac{y x^{-k}}{\sqrt{b^2}}\right )\right )\right \}\right \}$

Maple

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

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#### 41.30 problem number 30

problem number 366

Problem 2.3.2.30 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x +\left ( a x^{2 n} e^{\lambda x} y^2 + (b x^n e^{\lambda x} - n) y + c e^{\lambda x} \right ) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{-b^2 c (-\lambda x)^{-n} \sqrt{\frac{a x^{2 n}}{c}} \text{Gamma}(n,-\lambda x)+4 a c^2 (-\lambda x)^{-n} \sqrt{\frac{a x^{2 n}}{c}} \text{Gamma}(n,-\lambda x)-2 \sqrt{a} \sqrt{c} \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{\frac{b^2}{a c}} \sqrt{4 a c-b^2}-2 \sqrt{a} \sqrt{c} y \sqrt{4 a c-b^2} \sqrt{\frac{a x^{2 n}}{c}}}{4 a c-b^2}\right )}{4 a c-b^2}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{b}{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }} \left ( \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n,-\lambda \,x \right ) \sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}- \left ( -\lambda \,x \right ) ^{-n}\Gamma \left ( n \right ) \sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }{x}^{n}+2\,b\arctan \left ({\frac{b \left ( 2\,a{x}^{n}y+b \right ) }{\sqrt{{b}^{2} \left ( 4\,ca-{b}^{2} \right ) }}} \right ) \right ) } \right )$

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#### 41.31 problem number 31

problem number 367

Problem 2.3.2.31 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$y w_x + e^{\lambda x} \left ( (2 a \lambda x+a + b)y - e^{\lambda x}(a^2 \lambda x^2 + a b x -c) \right ) w_y = 0$

Mathematica

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

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac{1}{a} \left ( 2\,\lambda \,xa{{\rm e}^{2\,{1\arctan \left ({\frac{2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,\lambda \,xa-b}{a}{\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ){\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}}+\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}\int ^{-2\,{1\arctan \left ({\frac{2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,\lambda \,xa-b}{a}{\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ){\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}\!\tan \left ( 1/2\,{\it \_a}\,\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}} \right ){{\rm e}^{-{\it \_a}}}{d{\it \_a}}a+b{{\rm e}^{2\,{1\arctan \left ({\frac{2\,y\lambda \,{{\rm e}^{-\lambda \,x}}-2\,\lambda \,xa-b}{a}{\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}} \right ){\frac{1}{\sqrt{-{\frac{{b}^{2}+4\,\lambda \,c}{{a}^{2}}}}}}}}} \right ) } \right )$

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#### 41.32 problem number 32

problem number 368

Problem 2.3.2.32 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a e^{\lambda x} w_x + b y^m w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (-\frac{e^{-\lambda x} y^{-m} \left (a \lambda y e^{\lambda x}+b y^m-b m y^m\right )}{a \lambda (m-1)}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{{y}^{-m+1}a\lambda -b{{\rm e}^{-\lambda \,x}}m+b{{\rm e}^{-\lambda \,x}}}{a\lambda }} \right )$

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#### 41.33 problem number 33

problem number 369

Problem 2.3.2.33 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(a e^y + b x) w_x + w_y = 0$

Mathematica

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

Maple

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

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#### 41.34 problem number 34

problem number 370

Problem 2.3.2.34 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(a x^n e^{\lambda y} + b x y^m) w_x + e^{\mu y} w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({{x}^{ \left ( m+1 \right ) ^{-1}}{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}}{x}^{{\frac{m}{m+1}}} \left ({x}^{{\frac{mn}{m+1}}} \right ) ^{-1} \left ({x}^{{\frac{n}{m+1}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) }{\mu \, \left ( m+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) +\lambda \,\mu \,ym-{\mu }^{2}ym+\lambda \,\mu \,y-{\mu }^{2}y}{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y-a\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) -b{{\rm e}^{-1/2\,\mu \,y}}{y}^{m} \left ( \mu \,y \right ) ^{-m/2} \WhittakerM \left ( m/2,m/2+1/2,\mu \,y \right ) +\lambda \,\mu \,ym-{\mu }^{2}ym+\lambda \,\mu \,y-{\mu }^{2}y}{\mu \, \left ( m+1 \right ) }}}}\,{\rm d}y \right )$

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#### 41.35 problem number 35

problem number 371

Problem 2.3.2.35 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(a x^n y^m+ b x e^{\lambda y}) w_x + y^k w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({\frac{x}{{x}^{n}}{{\rm e}^{{\frac{{y}^{-k}{{\rm e}^{y\lambda }}bn}{\lambda }}}}{{\rm e}^{{\frac{ \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) bk{y}^{-k}}{\lambda }}}}{{\rm e}^{{\frac{{y}^{-k} \left ( -y\lambda \right ) ^{k}b\Gamma \left ( 1-k \right ) }{\lambda }}}} \left ({{\rm e}^{{\frac{ \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) bkn{y}^{-k}}{\lambda }}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{{y}^{-k} \left ( -y\lambda \right ) ^{k}b\Gamma \left ( 1-k \right ) n}{\lambda }}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{{y}^{-k}{{\rm e}^{y\lambda }}b}{\lambda }}}} \right ) ^{-1}}+an\int \!{y}^{-k+m}{{\rm e}^{{\frac{b{y}^{-k} \left ( n-1 \right ) \left ( k\Gamma \left ( -k \right ) \left ( -y\lambda \right ) ^{k}-k \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) +{{\rm e}^{y\lambda }} \right ) }{\lambda }}}}\,{\rm d}y-a\int \!{y}^{-k+m}{{\rm e}^{{\frac{b{y}^{-k} \left ( n-1 \right ) \left ( k\Gamma \left ( -k \right ) \left ( -y\lambda \right ) ^{k}-k \left ( -y\lambda \right ) ^{k}\Gamma \left ( -k,-y\lambda \right ) +{{\rm e}^{y\lambda }} \right ) }{\lambda }}}}\,{\rm d}y \right )$

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#### 41.36 problem number 36

problem number 372

Problem 2.3.2.36 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(a x^n y^m+ b x y^k) w_x + e^{\lambda y} w_y = 0$

Mathematica

$\text{Timed out}$ Timed out

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ({{x}^{ \left ( k+1 \right ) ^{-1}}{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}}{x}^{{\frac{k}{k+1}}} \left ({x}^{{\frac{kn}{k+1}}} \right ) ^{-1} \left ({x}^{{\frac{n}{k+1}}} \right ) ^{-1} \left ({{\rm e}^{{\frac{b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) }{\lambda \, \left ( k+1 \right ) }}}} \right ) ^{-1}}+an\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -{\lambda }^{2}yk-{\lambda }^{2}y}{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y-a\int \!{{\rm e}^{{\frac{bn{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -b{{\rm e}^{-1/2\,y\lambda }}{y}^{k} \left ( y\lambda \right ) ^{-k/2} \WhittakerM \left ( k/2,k/2+1/2,y\lambda \right ) -{\lambda }^{2}yk-{\lambda }^{2}y}{\lambda \, \left ( k+1 \right ) }}}}{y}^{m}\,{\rm d}y \right )$