124 HFOPDE, chapter 5.2.3

 124.1 Problem 1
 124.2 Problem 2
 124.3 Problem 3
 124.4 Problem 4
 124.5 Problem 5
 124.6 Problem 6
 124.7 Problem 7

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124.1 Problem 1

problem number 1005

Added March 12, 2019.

Problem Chapter 5.2.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = \alpha y w + \beta \sqrt{x y} + \gamma \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) =-{\frac{1}{b\alpha \, \left ( a+b \right ) y \left ( 5\,b+a \right ) \left ( 3\,b+a \right ) a}{{\rm e}^{1/2\,{\frac{\alpha \,y}{b}}}} \left ( -4\,a\sqrt{xy} \left ({\frac{\alpha \,y}{b}} \right ) ^{-1/4\,{\frac{3\,b+a}{b}}}{b}^{3}\beta \, \left ( 2\,\alpha \,y+a+3\,b \right ) \WhittakerM \left ( 1/4\,{\frac{-b+a}{b}},1/4\,{\frac{5\,b+a}{b}},{\frac{\alpha \,y}{b}} \right ) + \left ( 3\,b+a \right ) \left ( -2\,a\sqrt{xy} \left ({\frac{\alpha \,y}{b}} \right ) ^{-1/4\,{\frac{3\,b+a}{b}}}{b}^{2}\beta \, \left ( 3\,b+a \right ) \WhittakerM \left ( 1/4\,{\frac{3\,b+a}{b}},1/4\,{\frac{5\,b+a}{b}},{\frac{\alpha \,y}{b}} \right ) +\alpha \,{{\rm e}^{1/2\,{\frac{\alpha \,y}{b}}}} \left ( -a\gamma \,\ln \left ({\frac{\alpha \,y}{b}{x}^{-{\frac{b}{a}}}} \right ) -ab{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ) +\gamma \, \left ( a\ln \left ({\frac{\alpha \,y}{b}} \right ) +a\Ei \left ( 1,{\frac{\alpha \,y}{b}} \right ) -b\ln \left ( x \right ) \right ) \right ) y \left ( a+b \right ) \left ( 5\,b+a \right ) \right ) \right ) } \]

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124.2 Problem 2

problem number 1006

Added March 12, 2019.

Problem Chapter 5.2.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = \lambda \sqrt{x y} w + \beta x y + \gamma \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) =1/2\,{\frac{1}{{\lambda }^{2} \left ( a+b \right ) } \left ( -4\,\gamma \,\Ei \left ( 1,2\,{\frac{\sqrt{xy}\lambda }{a+b}} \right ){\lambda }^{2}{{\rm e}^{2\,{\frac{\sqrt{xy}\lambda }{a+b}}}}- \left ( -2\,{\it \_F1} \left ( y{x}^{-{\frac{b}{a}}} \right ){\lambda }^{2}{{\rm e}^{2\,{\frac{\sqrt{xy}\lambda }{a+b}}}}+\beta \, \left ( 2\,\sqrt{xy}\lambda +a+b \right ) \right ) \left ( a+b \right ) \right ) } \]

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124.3 Problem 3

problem number 1007

Added March 12, 2019.

Problem Chapter 5.2.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b x w_y = \alpha w + \beta \sqrt{x} + \gamma \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{\beta \,\sqrt{{\it \_a}}+\gamma }{\sqrt{a \left ({{\it \_a}}^{2}b+a{y}^{2}-{x}^{2}b \right ) }} \left ({\frac{ab{\it \_a}+\sqrt{a \left ({{\it \_a}}^{2}b+a{y}^{2}-{x}^{2}b \right ) }\sqrt{ab}}{\sqrt{ab}}} \right ) ^{-{\frac{\alpha }{\sqrt{ab}}}}}{d{\it \_a}}+{\it \_F1} \left ({\frac{a{y}^{2}-{x}^{2}b}{a}} \right ) \right ) \left ({\frac{abx}{\sqrt{ab}}}+\sqrt{ab{x}^{2}+ \left ( a{y}^{2}-{x}^{2}b \right ) a} \right ) ^{{\frac{\alpha }{\sqrt{ab}}}} \]

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124.4 Problem 4

problem number 1008

Added March 12, 2019.

Problem Chapter 5.2.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y w_x + b x w_y = \alpha w + \beta \sqrt{x} + \gamma \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac{\beta \,\sqrt{{\it \_a}}+\gamma }{\sqrt{a \left ( a{y}^{2}+ \left ({{\it \_a}}^{2}-{x}^{2} \right ) b \right ) }} \left ({\frac{ab{\it \_a}+\sqrt{a \left ( a{y}^{2}+ \left ({{\it \_a}}^{2}-{x}^{2} \right ) b \right ) }\sqrt{ab}}{\sqrt{ab}}} \right ) ^{-{\frac{\alpha }{\sqrt{ab}}}}}{d{\it \_a}}+{\it \_F1} \left ({\frac{a{y}^{2}-{x}^{2}b}{a}} \right ) \right ) \left ({\frac{abx}{\sqrt{ab}}}+\sqrt{{a}^{2}{y}^{2}} \right ) ^{{\frac{\alpha }{\sqrt{ab}}}} \]

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124.5 Problem 5

problem number 1009

Added March 12, 2019.

Problem Chapter 5.2.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \sqrt{x} w_x + b \sqrt{y} w_y = \alpha w + \beta x + \gamma y + \delta \]

Mathematica

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

Maple

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

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124.6 Problem 6

problem number 1010

Added March 12, 2019.

Problem Chapter 5.2.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \sqrt{x} w_x + b \sqrt{y} w_y = \alpha w + \beta \sqrt{x} + \gamma \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{y}\!{\frac{1}{b\sqrt{{\it \_a}}}{{\rm e}^{-2\,{\frac{\sqrt{{\it \_a}}\alpha }{b}}}} \left ( \gamma \,{\it \_a}+\beta \,\sqrt{{\frac{ \left ( \sqrt{{\it \_a}}a-\sqrt{y}a+b\sqrt{x} \right ) ^{2}}{{b}^{2}}}}+\delta \right ) }{d{\it \_a}}+{\it \_F1} \left ({\frac{-\sqrt{y}a+b\sqrt{x}}{b}} \right ) \right ){{\rm e}^{2\,{\frac{\sqrt{y}\alpha }{b}}}} \]

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124.7 Problem 7

problem number 1011

Added March 12, 2019.

Problem Chapter 5.2.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \sqrt{y} w_x + b \sqrt{x} w_y = \alpha w + \beta \sqrt{x} + \gamma \]

Mathematica

\[ \text{Timed out} \] Timed out

Maple

\[ w \left ( x,y \right ) = \left ( \int ^{y}\!{\frac{1}{b}{{\rm e}^{-{\frac{\alpha }{b}\int \!{\frac{1}{\sqrt{{\frac{ \left ( \left ({{\it \_b}}^{3/2}a+\RootOf \left ( x{b}^{2}- \left ({b}^{2}{y}^{3/2}a+{b}^{3}{\it \_Z} \right ) ^{2/3} \right ) b \right ){b}^{2} \right ) ^{2/3}}{{b}^{2}}}}}}\,{\rm d}{\it \_b}}}} \left ( \gamma \,{\it \_b}+\beta \,\sqrt{{\frac{ \left ( \left ({{\it \_b}}^{3/2}a+\RootOf \left ( x{b}^{2}- \left ({b}^{2}{y}^{3/2}a+{b}^{3}{\it \_Z} \right ) ^{2/3} \right ) b \right ){b}^{2} \right ) ^{2/3}}{{b}^{2}}}}+\delta \right ){\frac{1}{\sqrt{{\frac{ \left ( \left ({{\it \_b}}^{3/2}a+\RootOf \left ( x{b}^{2}- \left ({b}^{2}{y}^{3/2}a+{b}^{3}{\it \_Z} \right ) ^{2/3} \right ) b \right ){b}^{2} \right ) ^{2/3}}{{b}^{2}}}}}}}{d{\it \_b}}+{\it \_F1} \left ( \RootOf \left ( x{b}^{2}- \left ({b}^{2}{y}^{3/2}a+{b}^{3}{\it \_Z} \right ) ^{2/3} \right ) \right ) \right ){{\rm e}^{\int ^{y}\!{\frac{\alpha }{b}{\frac{1}{\sqrt{{\frac{ \left ({b}^{2}{{\it \_a}}^{3/2}a+{b}^{3}\RootOf \left ( x{b}^{2}- \left ({b}^{2}{y}^{3/2}a+{b}^{3}{\it \_Z} \right ) ^{2/3} \right ) \right ) ^{2/3}}{{b}^{2}}}}}}}{d{\it \_a}}}} \] contains RootOf