6.4.10 4.3

   6.4.10.1 [1087] Problem 1
   6.4.10.2 [1088] Problem 2
   6.4.10.3 [1089] Problem 3
   6.4.10.4 [1090] Problem 4
   6.4.10.5 [1091] Problem 5

6.4.10.1 [1087] Problem 1

problem number 1087

Added Feb. 23, 2019.

Problem Chapter 4.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c \tanh (\lambda x) + k \tanh (\mu y)) w \]

Mathematica

\[\left \{\left \{w(x,y)\to \cosh ^{\frac{c}{a \lambda }}(\lambda x) \cosh ^{\frac{k}{b \mu }}(\mu y) c_1\left (y-\frac{b x}{a}\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \left ( \tanh \left ( \lambda \,x \right ) -1 \right ) ^{-1/2\,{\frac{c}{a\lambda }}} \left ( \tanh \left ( \lambda \,x \right ) +1 \right ) ^{-1/2\,{\frac{c}{a\lambda }}} \left ( \tanh \left ( \mu \,y \right ) -1 \right ) ^{-1/2\,{\frac{k}{b\mu }}} \left ( \tanh \left ( \mu \,y \right ) +1 \right ) ^{-1/2\,{\frac{k}{b\mu }}}\]

________________________________________________________________________________________

6.4.10.2 [1088] Problem 2

problem number 1088

Added Feb. 23, 2019.

Problem Chapter 4.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c \tanh (\lambda x +\mu y) w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ya-bx}{a}} \right ) \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) -1 \right ) ^{-{\frac{c}{2\,a\lambda +2\,b\mu }}} \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) +1 \right ) ^{-{\frac{c}{2\,a\lambda +2\,b\mu }}}\]

________________________________________________________________________________________

6.4.10.3 [1089] Problem 3

problem number 1089

Added Feb. 23, 2019.

Problem Chapter 4.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \tanh (\lambda x +\mu y) w \]

Mathematica

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{y}{x}} \right ) \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) -1 \right ) ^{-1/2\,{a \left ({\frac{\mu \,y}{x}}+\lambda \right ) ^{-1}}} \left ( \tanh \left ( \lambda \,x+\mu \,y \right ) +1 \right ) ^{-1/2\,{a \left ({\frac{\mu \,y}{x}}+\lambda \right ) ^{-1}}}\]

________________________________________________________________________________________

6.4.10.4 [1090] Problem 4

problem number 1090

Added Feb. 23, 2019.

Problem Chapter 4.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b \tanh ^n(\lambda x) w_y = (c \tanh ^m(\mu x)+s \tanh ^k(\beta y)) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac{b \tanh ^{n+1}(\lambda x) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\tanh ^2(\lambda x)\right )}{a \lambda n+a \lambda }\right ) \exp \left (\int _1^x\frac{s \tanh ^k\left (\frac{\beta \left (-b \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\tanh ^2(\lambda x)\right ) \tanh ^{n+1}(\lambda x)+b \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};\tanh ^2(\lambda K[1])\right ) \tanh ^{n+1}(\lambda K[1])+a \lambda (n+1) y\right )}{a \lambda (n+1)}\right )+c \tanh ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac{b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ){{\rm e}^{\int ^{x}\!{\frac{1}{a} \left ( c \left ( \tanh \left ( \mu \,{\it \_b} \right ) \right ) ^{m}+s \left ( -\tanh \left ( \beta \, \left ( -\int \!{\frac{b \left ( \tanh \left ({\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}+\int \!{\frac{b \left ( \tanh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x-y \right ) \right ) \right ) ^{k} \right ) }{d{\it \_b}}}}\]

________________________________________________________________________________________

6.4.10.5 [1091] Problem 5

problem number 1091

Added Feb. 23, 2019.

Problem Chapter 4.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b \tanh ^n(\lambda y) w_y = (c \tanh ^m(\mu x)+s \tanh ^k(\beta y)) w \]

Mathematica

\[\left \{\left \{w(x,y)\to c_1\left (\frac{\tanh ^{1-n}(\lambda y) \, _2F_1\left (1,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};\tanh ^2(\lambda y)\right )}{\lambda -\lambda n}-\frac{b x}{a}\right ) \exp \left (\int _1^y\frac{\tanh ^{-n}(\lambda K[1]) \left (s \tanh ^k(\beta K[1])+c \tanh ^m\left (\frac{-a \mu \, _2F_1\left (1,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};\tanh ^2(\lambda y)\right ) \tanh ^{1-n}(\lambda y)+a \mu \, _2F_1\left (1,\frac{1}{2}-\frac{n}{2};\frac{3}{2}-\frac{n}{2};\tanh ^2(\lambda K[1])\right ) \tanh ^{1-n}(\lambda K[1])+b \lambda \mu x-b \lambda \mu n x}{b \lambda -b \lambda n}\right )\right )}{b}dK[1]\right )\right \}\right \}\]

Maple

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

________________________________________________________________________________________