### 93 HFOPDE, chapter 3.8.3

93.1 Problem 1
93.2 Problem 2
93.3 Problem 3
93.4 Problem 4
93.5 Problem 5
93.6 Problem 6
93.7 Problem 7
93.8 Problem 8

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#### 93.1 Problem 1

problem number 798

Problem Chapter 3.8.3.1 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a w_x + b w_y = f(\alpha x+\beta y)$

Mathematica

$\left \{\left \{w(x,y)\to \int _1^x \frac{f\left (\frac{\beta (b K[1]+a y-b x)}{a}+\alpha K[1]\right )}{a} \, dK[1]+c_1\left (\frac{a y-b x}{a}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{a}f \left ({\frac{ \left ( ay-bx \right ) \beta +{\it \_a}\,a\alpha +b\beta \,{\it \_a}}{a}} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{ay-bx}{a}} \right )$

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#### 93.2 Problem 2

problem number 799

Problem Chapter 3.8.3.2 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + y w_y = x f(\frac{y}{x})$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\frac{y}{x}\right )+x f\left (\frac{y}{x}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) =xf \left ({\frac{y}{x}} \right ) +{\it \_F1} \left ({\frac{y}{x}} \right )$

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#### 93.3 Problem 3

problem number 800

Problem Chapter 3.8.3.2 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + y w_y = f(x^2+y^2)$

Mathematica

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

Maple

$w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{{\it \_a}}f \left ({\frac{{y}^{2}{{\it \_a}}^{2}}{{x}^{2}}}+{{\it \_a}}^{2} \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{y}{x}} \right )$

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#### 93.4 Problem 4

problem number 801

Problem Chapter 3.8.3.4 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x w_x + y w_y = x f(\frac{y}{x})+ g(x^2+y^2)$

Mathematica

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

Maple

$w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{{\it \_a}} \left ({\it \_a}\,f \left ({\frac{y}{x}} \right ) +g \left ({\frac{{y}^{2}{{\it \_a}}^{2}}{{x}^{2}}}+{{\it \_a}}^{2} \right ) \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{y}{x}} \right )$

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#### 93.5 Problem 5

problem number 802

Problem Chapter 3.8.3.5 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$a x w_x + b y w_y = x^k f(x^n y^m)$

Mathematica

$\left \{\left \{w(x,y)\to \int _1^x \frac{K[1]^{k-1} f\left (K[1]^{m+n}\right )}{a} \, dK[1]+c_1\left (y x^{-\frac{b}{a}}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) =\int ^{x}\!{\frac{{{\it \_a}}^{k-1}}{a}f \left ({{\it \_a}}^{n} \left ( y{x}^{-{\frac{b}{a}}}{{\it \_a}}^{{\frac{b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right )$

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#### 93.6 Problem 6

problem number 803

Problem Chapter 3.8.3.6 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$m x w_x + n y w_y = f(a x^n + b y^m)$

Mathematica

$\left \{\left \{w(x,y)\to \int _1^x \frac{f\left (a K[1]^n+b K[1]^m\right )}{m K[1]} \, dK[1]+c_1\left (y x^{-\frac{n}{m}}\right )\right \}\right \}$

Maple

$w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{m{\it \_a}}f \left ( a{{\it \_a}}^{n}+b \left ( y{x}^{-{\frac{n}{m}}}{{\it \_a}}^{{\frac{n}{m}}} \right ) ^{m} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac{n}{m}}} \right )$

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#### 93.7 Problem 7

problem number 804

Problem Chapter 3.8.3.7 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$x^2 w_x + x y w_y = y^k f(\alpha x + \beta y)$

Mathematica

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

Maple

$w \left ( x,y \right ) =\int ^{x}\!{\frac{1}{{{\it \_a}}^{2}}f \left ({\it \_a}\, \left ({\frac{\beta \,y}{x}}+\alpha \right ) \right ) \left ({\frac{y{\it \_a}}{x}} \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ({\frac{y}{x}} \right )$

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#### 93.8 Problem 8

problem number 805

Problem Chapter 3.8.3.8 from Handbook of ﬁrst order partial diﬀerential equations by Polyanin, Zaitsev, Moussiaux.

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

$\frac{f(x)}{f'(x)} w_x + \frac{g(y)}{g'(y)} w_y = h(f(x)+g(y))$

Mathematica

$\left \{\left \{w(x,y)\to c_1\left (\log \left (\frac{\text{InverseFunction}\left [g^{(-1)},1,1\right ][y]}{f(x)}\right )\right )+\int _1^x \frac{f'(K[1]) h\left (g\left (\text{InverseFunction}\left [\text{InverseFunction}\left [g^{(-1)},1,1\right ],1,1\right ]\left [\frac{f(K[1]) \text{InverseFunction}\left [g^{(-1)},1,1\right ][y]}{f(x)}\right ]\right )+f(K[1])\right )}{f(K[1])} \, dK[1]\right \}\right \}$

Maple

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