#### 6.4.8 4.1

6.4.8.1 [1077] Problem 1
6.4.8.2 [1078] Problem 2
6.4.8.3 [1079] Problem 3
6.4.8.4 [1080] Problem 4
6.4.8.5 [1081] Problem 5

##### 6.4.8.1 [1077] Problem 1

problem number 1077

Problem Chapter 4.4.1.1, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y)$$ $a w_x + b w_y = (c \sinh (\lambda x) + k \sinh (\mu y)) w$

Mathematica

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

Maple

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

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##### 6.4.8.2 [1078] Problem 2

problem number 1078

Problem Chapter 4.4.1.2, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y)$$ $a w_x + b w_y = c \sinh (\lambda x +\mu y) w$

Mathematica

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

Maple

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

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##### 6.4.8.3 [1079] Problem 3

problem number 1079

Problem Chapter 4.4.1.3, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $$w(x,y)$$ $x w_x + y w_y = a x \sinh (\lambda x +\mu y) w$

Mathematica

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

Maple

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

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##### 6.4.8.4 [1080] Problem 4

problem number 1080

Problem Chapter 4.4.1.4, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

\$Aborted

Maple

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

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##### 6.4.8.5 [1081] Problem 5

problem number 1081

Problem Chapter 4.4.1.5, from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica

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

Maple

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

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