### 64 HFOPDE, chapter 2.9.1

64.1 problem number 1
64.2 problem number 2
64.3 problem number 3
64.4 problem number 4
64.5 problem number 5

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#### 64.1 problem number 1

problem number 592

Problem 2.9.1.1 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$f(x) w_x + g(y) w_y = 0$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\int _1^y \frac{1}{g(K[1])} \, dK[1]-\int _1^x \frac{1}{f(K[2])} \, dK[2]\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right )$

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#### 64.2 problem number 2

problem number 593

Problem 2.9.1.2 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(f(x)+g(y)) w_x + f'(x) w_y = 0$

Mathematica

$\text{DSolve}\left [f'(x) w^{(0,1)}(x,y)+w^{(1,0)}(x,y) (f(x)+g(y))=0,w(x,y),\{x,y\}\right ]$

Maple

$w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{{\rm e}^{-y}}g \left ( y \right ) \,{\rm d}y+{{\rm e}^{-y}}f \left ( x \right ) \right )$

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#### 64.3 problem number 3

problem number 594

Problem 2.9.1.3 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(x^n f(y) + x g(y)) w_x + h(y) w_y = 0$

Mathematica

$\text{DSolve}\left [w^{(1,0)}(x,y) \left (f(y) x^n+x g(y)\right )+h(y) w^{(0,1)}(x,y)=0,w(x,y),\{x,y\}\right ]$

Maple

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

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#### 64.4 problem number 4

problem number 595

Problem 2.9.1.4 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(f(y) + a m x^n y^{m-1}) w_x - (g(x)+a n x^{n-1} y^m) w_y = 0$

Mathematica

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

Maple

$\text{ sol=() }$

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#### 64.5 problem number 5

problem number 596

Problem 2.9.1.5 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$(e^{\alpha x} f(y) + c \beta ) w_x - (e^{\beta y} g(x) + c \alpha ) w_y = 0$

Mathematica

$\text{DSolve}\left [w^{(1,0)}(x,y) \left (e^{\alpha x} f(y)+\beta c\right )-w^{(0,1)}(x,y) \left (\alpha c+e^{\beta y} g(x)\right )=0,w(x,y),\{x,y\}\right ]$

Maple

$\text{ sol=() }$