98 HFOPDE, chapter 4.2.3

 98.1 Problem 1
 98.2 Problem 2
 98.3 Problem 3
 98.4 Problem 4
 98.5 Problem 5
 98.6 Problem 6

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98.1 Problem 1

problem number 831

Added Feb. 17, 2019.

Problem Chapter 4.2.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = (c x^3+d y^3) w \]

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{a y-b x}{a}\right ) e^{\frac{c x^4}{4 a}+\frac{d y^4}{4 b}}\right \}\right \} \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{ay-bx}{a}} \right ){{\rm e}^{{\frac{ \left ( ay-bx \right ) ^{3}dx}{{a}^{4}}}+3/2\,{\frac{ \left ( ay-bx \right ) ^{2}bd{x}^{2}}{{a}^{4}}}+{\frac{ \left ( ay-bx \right ){b}^{2}d{x}^{3}}{{a}^{4}}}+1/4\,{\frac{c{x}^{4}}{a}}+1/4\,{\frac{{b}^{3}d{x}^{4}}{{a}^{4}}}}} \]

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98.2 Problem 2

problem number 832

Added Feb. 17, 2019.

Problem Chapter 4.2.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a \sqrt{x^2+y^2} w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ({\frac{y}{x}} \right ){{\rm e}^{a\sqrt{{x}^{2}+{y}^{2}}}} \]

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98.3 Problem 3

problem number 833

Added Feb. 17, 2019.

Problem Chapter 4.2.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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98.4 Problem 4

problem number 834

Added Feb. 17, 2019.

Problem Chapter 4.2.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={x}^{b}{\it \_F1} \left ( y{x}^{-a} \right ){{\rm e}^{-{\frac{da}{x \left ( a+1 \right ) }}-{\frac{d}{x \left ( a+1 \right ) }}-{\frac{c}{y \left ( a+1 \right ) }}-{\frac{k}{yx \left ( a+1 \right ) }}-{\frac{c}{ay \left ( a+1 \right ) }}}} \]

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98.5 Problem 5

problem number 835

Added Feb. 17, 2019.

Problem Chapter 4.2.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x y^2 w_x + b x^2 y w_y = (a n y^2+ b m x^2) w \]

Mathematica

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

Maple

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

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98.6 Problem 6

problem number 836

Added Feb. 17, 2019.

Problem Chapter 4.2.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^3 w_x + a y^3 w_y = x^2 (b x + c y) w \]

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{{y}^{2}a-{x}^{2}}{{x}^{2}{y}^{2}}} \right ){{\rm e}^{bx}} \left ( x\sqrt{-{\frac{{y}^{2}a-{x}^{2}}{{x}^{2}{y}^{2}}}}+\sqrt{-{\frac{{y}^{2}a-{x}^{2}}{{y}^{2}}}+a} \right ) ^{{c{\frac{1}{\sqrt{-{\frac{{y}^{2}a-{x}^{2}}{{x}^{2}{y}^{2}}}}}}}} \]