69 HFOPDE, chapter 3.2.2

 69.1 Problem 1
 69.2 Problem 2
 69.3 Problem 3
 69.4 Problem 4
 69.5 Problem 5
 69.6 Problem 6
 69.7 Problem 7

_______________________________________________________________________________________

69.1 Problem 1

problem number 646

Added Feb. 9, 2019.

Problem Chapter 3.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c x^2+d y^2+ k x y+n \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{6 a^3 c_1\left (\frac{a y-b x}{a}\right )+2 a^2 c x^3+6 a^2 d x y^2+3 a^2 k x^2 y+6 a^2 n x-6 a b d x^2 y-a b k x^3+2 b^2 d x^3}{6 a^3}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =1/6\,{\frac{ \left ( 2\,{a}^{2}c-abk+2\,{b}^{2}d \right ){x}^{3}}{{a}^{3}}}+1/6\,{\frac{ \left ( 3\,{a}^{2}k-6\,abd \right ) y{x}^{2}}{{a}^{3}}}+ \left ({\frac{d{y}^{2}}{a}}+{\frac{n}{a}} \right ) x+{\it \_F1} \left ({\frac{ay-bx}{a}} \right ) \]

_______________________________________________________________________________________

69.2 Problem 2

problem number 647

Added Feb. 9, 2019.

Problem Chapter 3.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c x^2+d y^2+ k x y+n \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\frac{kxy}{a+b}}+1/2\,{\frac{c{x}^{2}}{a}}+{\frac{n\ln \left ( x \right ) }{a}}+1/2\,{\frac{d{y}^{2}}{b}}+{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ) \]

_______________________________________________________________________________________

69.3 Problem 3

problem number 648

Added Feb. 9, 2019.

Problem Chapter 3.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

_______________________________________________________________________________________

69.4 Problem 4

problem number 649

Added Feb. 9, 2019.

Problem Chapter 3.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^2 w_x + b y^2 w_y = c x^2+d y^2+ k x y+ n x+ m y+s \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a^2 b x^2 c_1\left (\frac{b y-a x}{a x y}\right )-a^2 m x^2 \log \left (b-\frac{b y-a x}{y}\right )+a^2 m x^2 \log (x)-a b^2 x y c_1\left (\frac{b y-a x}{a x y}\right )+a b c x^3-a b d x y^2+a b k x^2 y \log \left (b-\frac{b y-a x}{y}\right )-a b m x y \log (x)+a b m x y \log \left (b-\frac{b y-a x}{y}\right )+a b n x^2 \log (x)-a b s x-b^2 c x^2 y-b^2 n x y \log (x)+b^2 s y}{a b x (a x-b y)}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{cx}{a}}-{\frac{s}{ax}}+{\frac{m\ln \left ( x \right ) }{b}}+{\frac{n\ln \left ( x \right ) }{a}}+{\frac{daxy}{-ax+by} \left ( -{\frac{-ax+by}{y}}+b \right ) ^{-1}}-{\frac{m}{b}\ln \left ( -{\frac{-ax+by}{y}}+b \right ) }-{\frac{kxy}{-ax+by}\ln \left ( -{\frac{-ax+by}{y}}+b \right ) }+{\it \_F1} \left ( -{\frac{-ax+by}{axy}} \right ) \]

_______________________________________________________________________________________

69.5 Problem 5

problem number 650

Added Feb. 9, 2019.

Problem Chapter 3.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + a x y w_y = b y^2 \]

Mathematica

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

Maple

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

_______________________________________________________________________________________

69.6 Problem 6

problem number 651

Added Feb. 9, 2019.

Problem Chapter 3.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^2 w_x + b x^2 w_y = c x^2+d \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{b d x \left (\frac{a y^3}{a y^3-b x^3}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{4}{3},-\frac{b x^3}{a y^3-b x^3}\right )+\sqrt [3]{a} b \left (a y^3\right )^{2/3} c_1\left (\frac{a y^3-b x^3}{3 a}\right )+a c y^3}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}\right \},\left \{w(x,y)\to \frac{-\sqrt [3]{-1} b d x \left (\frac{a y^3}{a y^3-b x^3}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{4}{3},-\frac{b x^3}{a y^3-b x^3}\right )+\sqrt [3]{a} b \left (a y^3\right )^{2/3} c_1\left (\frac{a y^3-b x^3}{3 a}\right )-\sqrt [3]{-1} a c y^3}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}\right \},\left \{w(x,y)\to \frac{(-1)^{2/3} b d x \left (\frac{a y^3}{a y^3-b x^3}\right )^{2/3} \text{Hypergeometric2F1}\left (\frac{1}{3},\frac{2}{3},\frac{4}{3},-\frac{b x^3}{a y^3-b x^3}\right )+\sqrt [3]{a} b \left (a y^3\right )^{2/3} c_1\left (\frac{a y^3-b x^3}{3 a}\right )+(-1)^{2/3} a c y^3}{\sqrt [3]{a} b \left (a y^3\right )^{2/3}}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac{ \left ({{\it \_a}}^{2}c+d \right ) a}{ \left ( \left ({{\it \_a}}^{3}b+\RootOf \left ( ay-\sqrt [3]{{a}^{2}b{x}^{3}+{a}^{3}{\it \_Z}} \right ) a \right ){a}^{2} \right ) ^{2/3}}}{d{\it \_a}}+{\it \_F1} \left ( \RootOf \left ( ay-\sqrt [3]{{a}^{2}b{x}^{3}+{a}^{3}{\it \_Z}} \right ) \right ) \] Contains unresolved integral with RootOf

_______________________________________________________________________________________

69.7 Problem 7

problem number 652

Added Feb. 9, 2019.

Problem Chapter 3.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^2 w_x + b x y w_y = c x^2+d y^2 \]

Mathematica

\[ \left \{\left \{w(x,y)\to \frac{a^{3/2} (-c) \sqrt{\frac{a y^2-b x^2}{a}} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a} \sqrt{\frac{a y^2-b x^2}{a}}}\right )+a b^{3/2} c_1\left (\frac{a y^2-b x^2}{2 a}\right )+a \sqrt{b} c x+b^{3/2} d x}{a b^{3/2}}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\frac{c{x}^{2}}{\sqrt{b \left ({y}^{2}a-b{x}^{2} \right ) }}\arctan \left ({\frac{bx}{\sqrt{b \left ({y}^{2}a-b{x}^{2} \right ) }}} \right ) }+ \left ({\frac{c}{b}}+{\frac{d}{a}} \right ) x-{\frac{c{y}^{2}a}{b\sqrt{b \left ({y}^{2}a-b{x}^{2} \right ) }}\arctan \left ({\frac{bx}{\sqrt{b \left ({y}^{2}a-b{x}^{2} \right ) }}} \right ) }+{\it \_F1} \left ({\frac{{y}^{2}a-b{x}^{2}}{a}} \right ) \]