40 HFOPDE, chapter 2.3.1

 40.1 problem number 1
 40.2 problem number 2
 40.3 problem number 3
 40.4 problem number 4
 40.5 problem number 5
 40.6 problem number 6
 40.7 problem number 7
 40.8 problem number 8
 40.9 problem number 9
 40.10 problem number 10
 40.11 problem number 11

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40.1 problem number 1

problem number 326

Added January 2, 2019.

Problem 2.3.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a e^{\lambda x} w_y = 0 \]

Mathematica

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

Maple

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

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40.2 problem number 2

problem number 327

Added January 7, 2019.

Problem 2.3.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a e^{\lambda x} +b \right ) w_y = 0 \]

Mathematica

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

Maple

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

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40.3 problem number 3

problem number 328

Added January 7, 2019.

Problem 2.3.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a e^{\lambda y} +b \right ) w_y = 0 \]

Mathematica

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

Maple

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

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40.4 problem number 4

problem number 329

Added January 7, 2019.

Problem 2.3.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a e^{\lambda y+ \beta x} +b \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{-bx\lambda +y\lambda +\ln \left ( \left ( a\lambda \,{{\rm e}^{\beta \,x+y\lambda }}+\lambda \,b+\beta \right ) ^{-1} \right ) }{\lambda \,b+\beta }} \right ) \]

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40.5 problem number 5

problem number 330

Added January 7, 2019.

Problem 2.3.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a e^{\lambda y+ \beta x} +b e^{\gamma x}\right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{1}{\lambda } \left ({{\rm e}^{{\frac{\lambda \, \left ( b{{\rm e}^{gx}}-gy \right ) }{g}}}}+a\int \!{{\rm e}^{\beta \,x+{\frac{\lambda \,b{{\rm e}^{gx}}}{g}}}}\,{\rm d}x\lambda \right ) } \right ) \]

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40.6 problem number 6

problem number 331

Added January 7, 2019.

Problem 2.3.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a e^{\lambda x} w_x + b e^{\beta y} w_y = 0 \]

Mathematica

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

Maple

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

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40.7 problem number 7

problem number 332

Added January 7, 2019.

Problem 2.3.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a e^{\lambda x} +b \right ) w_x + \left ( c e^{\beta x}+d \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{-\lambda e^{\beta x} \text{Hypergeometric2F1}\left (1,\frac{\beta }{\lambda },\frac{\beta }{\lambda }+1,-\frac{a e^{\lambda x}}{b}\right )+\beta c \log \left (a e^{\lambda x}+b\right )+\beta d \log \left (a e^{\lambda x}+b\right )+b \beta \lambda y-\beta c \lambda x-\beta d \lambda x}{b \beta \lambda }\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac{c+{{\rm e}^{\beta \,x}}+d}{a{{\rm e}^{\lambda \,x}}+b}}\,{\rm d}x+y \right ) \]

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40.8 problem number 8

problem number 333

Added January 7, 2019.

Problem 2.3.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a e^{\lambda x} +b \right ) w_x + \left ( c e^{\beta y}+d \right ) w_y = 0 \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (-\frac{\log \left (\left (c e^{\frac{\beta c x}{b}+\frac{\beta d x}{b}-\beta y}+d e^{\frac{\beta c x}{b}+\frac{\beta d x}{b}-\beta y}+e^{\frac{\beta x (c+d)}{b}}\right ) \left (a e^{\lambda x}+b\right )^{-\frac{\beta c}{b \lambda }-\frac{\beta d}{b \lambda }}\right )}{\beta (c+d)}\right )\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{1}{\lambda \,b\beta \, \left ( c+d \right ) } \left ( yb\beta \,\lambda -\lambda \,x\beta \,c-\lambda \,x\beta \,d+\ln \left ( a{{\rm e}^{\lambda \,x}}+b \right ) \beta \,c+\ln \left ( a{{\rm e}^{\lambda \,x}}+b \right ) \beta \,d-\RootOf \left ({{\rm e}^{{\frac{c{\it \_Z}}{c+d}}}} \left ({{\rm e}^{{\frac{y\beta \,c}{c+d}}}} \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac{\beta \,{c}^{2}}{\lambda \,b \left ( c+d \right ) }}} \left ( \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac{\beta \,cd}{\lambda \,b \left ( c+d \right ) }}} \right ) ^{2}{{\rm e}^{{\frac{y\beta \,d}{c+d}}}} \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac{\beta \,{d}^{2}}{\lambda \,b \left ( c+d \right ) }}}-{{\rm e}^{{\frac{c{\it \_Z}}{c+d}}}}{{\rm e}^{{\frac{{\it \_Z}\,d}{c+d}}}} \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac{\beta \, \left ( c+d \right ) }{\lambda \,b}}}+ \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac{\beta \, \left ( c+d \right ) }{\lambda \,b}}}c+ \left ( a{{\rm e}^{\lambda \,x}}+b \right ) ^{{\frac{\beta \, \left ( c+d \right ) }{\lambda \,b}}}d \right ) \right ) \lambda \,b \right ) } \right ) \] Has RootOf

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40.9 problem number 9

problem number 334

Added January 7, 2019.

Problem 2.3.1.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a e^{\lambda y} +b \right ) w_x + \left ( c e^{\beta x}+d \right ) w_y = 0 \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{yb\beta \,\lambda -\lambda \,x\beta \,c-\lambda \,x\beta \,d+a{{\rm e}^{y\lambda }}\beta -{{\rm e}^{\beta \,x}}\lambda }{\beta \,\lambda }} \right ) \]

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40.10 problem number 10

problem number 335

Added January 7, 2019.

Problem 2.3.1.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a e^{\lambda x} +b e^{\beta y}\right ) w_x + a \lambda e^{\lambda x} w_y = 0 \]

Mathematica

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

Maple

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

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40.11 problem number 11

problem number 336

Added January 7, 2019.

Problem 2.3.1.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \left ( a e^{\lambda x+\beta y} +c \mu \right ) w_x - \left ( b e^{\gamma x+ mu y}+c \lambda \right ) w_y = 0 \]

Mathematica

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

Maple

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