### 76 HFOPDE, chapter 3.4.3

76.1 Problem 1
76.2 Problem 2
76.3 Problem 3
76.4 Problem 4
76.5 Problem 5

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#### 76.1 Problem 1

problem number 702

Problem Chapter 3.4.3.1 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = c \tanh (\lambda x)+ k \tanh (\mu y)$

Mathematica

$\left \{\left \{w(x,y)\to \frac{a b \lambda \mu c_1\left (\frac{a y-b x}{a}\right )+a k \lambda \log \left (\cosh \left (\frac{\mu (a y-b x)}{a}+\frac{b \mu x}{a}\right )\right )+b c \mu \log (\cosh (\lambda x))}{a b \lambda \mu }\right \}\right \}$

Maple

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

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#### 76.2 Problem 2

problem number 703

Problem Chapter 3.4.3.2 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = c \tanh (\lambda x+\mu y)$

Mathematica

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

Maple

$w \left ( x,y \right ) =1/2\,{\frac{1}{\lambda \,a+\mu \,b} \left ( 2\,{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) a\lambda +2\,{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) b\mu -c\ln \left ( \tanh \left ({\frac{\mu \, \left ( ay-bx \right ) +ax\lambda +\mu \,bx}{a}} \right ) -1 \right ) -c\ln \left ( \tanh \left ({\frac{\mu \, \left ( ay-bx \right ) +ax\lambda +\mu \,bx}{a}} \right ) +1 \right ) \right ) }$

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#### 76.3 Problem 3

problem number 704

Problem Chapter 3.4.3.3 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + y w_y = a x \tanh (\lambda x+\mu y)$

Mathematica

$\left \{\left \{w(x,y)\to \frac{a x \log \left (\cosh \left (x \left (\lambda +\frac{\mu y}{x}\right )\right )\right )+\lambda x c_1\left (\frac{y}{x}\right )+\mu y c_1\left (\frac{y}{x}\right )}{\lambda x+\mu y}\right \}\right \}$

Maple

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

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#### 76.4 Problem 4

problem number 705

Problem Chapter 3.4.3.4 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b \tanh ^n(\lambda x) w_y = c \tanh ^m(\mu x)+s \tanh ^k(\beta y)$

Mathematica

$\text{Timed out}$ Timed out

Maple

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

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#### 76.5 Problem 5

problem number 706

Problem Chapter 3.4.3.5 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b \tanh ^n(\lambda y) w_y = c \tanh ^m(\mu x)+s \tanh ^k(\beta y)$

Mathematica

$\text{Timed out}$ Timed out

Maple

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