68 HFOPDE, chapter 3.2.1

 68.1 Problem 1
 68.2 Problem 2
 68.3 Problem 3
 68.4 Problem 4
 68.5 Problem 5
 68.6 Problem 6
 68.7 Problem 7
 68.8 Problem 8

_______________________________________________________________________________________

68.1 Problem 1

problem number 638

Added Feb. 9, 2019.

Problem Chapter 3.2.1.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{cx}{a}}+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

_______________________________________________________________________________________

68.2 Problem 2

problem number 639

Added Feb. 9, 2019.

Problem Chapter 3.2.1.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{2 a^2 c_1\left (\frac{a y-b x}{a}\right )+a \alpha x^2+2 a \beta x y+2 a \gamma x-b \beta x^2}{2 a^2}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =1/2\,{\frac{ \left ( a\alpha -b\beta \right ){x}^{2}}{{a}^{2}}}+ \left ({\frac{\beta \,y}{a}}+{\frac{\gamma }{a}} \right ) x+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

_______________________________________________________________________________________

68.3 Problem 3

problem number 640

Added Feb. 9, 2019.

Problem Chapter 3.2.1.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{2 a^2 c_1\left (\frac{a y-b \log (x)}{a}\right )+2 a \alpha x+2 a \beta y \log (x)+2 a \gamma \log (x)-b \beta \log ^2(x)}{2 a^2}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{\alpha \,x}{a}}+{\frac{\ln \left ( x \right ) \beta \,y}{a}}-1/2\,{\frac{1}{{a}^{2}} \left ( b\beta \, \left ( \ln \left ( x \right ) \right ) ^{2}-2\,\gamma \,\ln \left ( x \right ) a-2\,{\it \_F1} \left ( -{\frac{b\ln \left ( x \right ) -ay}{a}} \right ){a}^{2} \right ) } \]

_______________________________________________________________________________________

68.4 Problem 4

problem number 641

Added Feb. 9, 2019.

Problem Chapter 3.2.1.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b x w_y = c \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{1}{a} \left ( c\ln \left ( x \right ) +{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) a \right ) } \]

_______________________________________________________________________________________

68.5 Problem 5

problem number 642

Added Feb. 9, 2019.

Problem Chapter 3.2.1.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x +b) w_x + (c y +d) w_y = \alpha x+ \beta y + \gamma \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a^2 c^2 c_1\left (\frac{(c y+d) (a x+b)^{-\frac{c}{a}}}{c}\right )+a^2 \beta c y+a^2 \beta d-\alpha b c^2 \log (a x+b)+a \alpha c^2 x-a \beta c d \log (a x+b)+a c^2 \gamma \log (a x+b)}{a^2 c^2}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{\alpha \,x}{a}}+{\frac{\beta \,y}{c}}+{\frac{1}{{a}^{2}{c}^{2}} \left ({\it \_F1} \left ({\frac{cy+d}{c} \left ( ax+b \right ) ^{-{\frac{c}{a}}}} \right ){a}^{2}{c}^{2}+\ln \left ( ax+b \right ) \gamma \,a{c}^{2}-\ln \left ( ax+b \right ) \beta \,dac-\ln \left ( ax+b \right ) \alpha \,b{c}^{2}+{a}^{2}\beta \,d \right ) } \]

_______________________________________________________________________________________

68.6 Problem 6

problem number 643

Added Feb. 9, 2019.

Problem Chapter 3.2.1.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b w_y = \alpha x+ \beta y + \gamma \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{-a^{3/2} \alpha y^2 \sqrt{a y^2}+3 \sqrt{a} \alpha b x \sqrt{a y^2}+3 a b^2 c_1\left (\frac{a y^2-2 b x}{2 a}\right )+3 \sqrt{a} b \gamma \sqrt{a y^2}+3 b^2 \beta x}{3 a b^2}\right \},\left \{w(x,y)\to \frac{a^{3/2} \alpha y^2 \sqrt{a y^2}-3 \sqrt{a} \alpha b x \sqrt{a y^2}+3 a b^2 c_1\left (\frac{a y^2-2 b x}{2 a}\right )-3 \sqrt{a} b \gamma \sqrt{a y^2}+3 b^2 \beta x}{3 a b^2}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =1/6\,{\frac{ \left ( 6\,\sqrt{ \left ({y}^{2}a-2\,bx \right ) a+2\,abx}a\alpha \,b+6\,a{b}^{2}\beta \right ) x}{{a}^{2}{b}^{2}}}-1/2\,{\frac{\sqrt{ \left ({y}^{2}a-2\,bx \right ) a+2\,abx}\alpha \,{y}^{2}}{{b}^{2}}}+1/6\,{\frac{1}{{a}^{2}{b}^{2}} \left ( 6\,{\it \_F1} \left ({\frac{{y}^{2}a-2\,bx}{a}} \right ){a}^{2}{b}^{2}+ \left ( \left ({y}^{2}a-2\,bx \right ) a+2\,abx \right ) ^{3/2}\alpha +6\,\gamma \,\sqrt{ \left ({y}^{2}a-2\,bx \right ) a+2\,abx}ab \right ) } \]

_______________________________________________________________________________________

68.7 Problem 7

problem number 644

Added Feb. 9, 2019.

Problem Chapter 3.2.1.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b x w_y = c \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{\sqrt{a} \sqrt{b} c_1\left (\frac{a y^2-b x^2}{2 a}\right )-c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a y^2}}\right )}{\sqrt{a} \sqrt{b}}\right \},\left \{w(x,y)\to \frac{\sqrt{a} \sqrt{b} c_1\left (\frac{a y^2-b x^2}{2 a}\right )+c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a y^2}}\right )}{\sqrt{a} \sqrt{b}}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{1}{\sqrt{ab}} \left ({\it \_F1} \left ({\frac{{y}^{2}a-b{x}^{2}}{a}} \right ) \sqrt{ab}+c\ln \left ({\frac{abx}{\sqrt{ab}}}+\sqrt{ab{x}^{2}+ \left ({y}^{2}a-b{x}^{2} \right ) a} \right ) \right ) } \]

_______________________________________________________________________________________

68.8 Problem 8

problem number 645

Added Feb. 9, 2019.

Problem Chapter 3.2.1.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b x w_y = c x+ k y \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a b c_1\left (\frac{a y^2-b x^2}{2 a}\right )-\sqrt{a} c \sqrt{a y^2}+b k x}{a b}\right \},\left \{w(x,y)\to \frac{a b c_1\left (\frac{a y^2-b x^2}{2 a}\right )+\sqrt{a} c \sqrt{a y^2}+b k x}{a b}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{cy}{b}}+{\frac{kx}{a}}+{\it \_F1} \left ({\frac{{y}^{2}a-b{x}^{2}}{a}} \right ) \]