97 HFOPDE, chapter 4.2.2

 97.1 Problem 1
 97.2 Problem 2
 97.3 Problem 3
 97.4 Problem 4
 97.5 Problem 5
 97.6 Problem 6
 97.7 Problem 7
 97.8 Problem 8

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97.1 Problem 1

problem number 823

Added Feb. 17, 2019.

Problem Chapter 4.2.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \left \{\left \{w(x,y)\to c_1\left (\frac{a y-b x}{a}\right ) \exp \left (-\frac{b^2 x^3}{3 a^3}-\frac{b x^2 (a y-b x)}{a^3}-\frac{x (a y-b x)^2}{a^3}+\frac{x^3}{3 a}\right )\right \}\right \} \]

Maple

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

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97.2 Problem 2

problem number 824

Added Feb. 17, 2019.

Problem Chapter 4.2.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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97.3 Problem 3

problem number 825

Added Feb. 17, 2019.

Problem Chapter 4.2.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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97.4 Problem 4

problem number 826

Added Feb. 17, 2019.

Problem Chapter 4.2.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac{ay-x}{yx}} \right ) \left ( -{\frac{ay-x}{y}}+a \right ) ^{-{\frac{cxy}{ay-x}}}{{\rm e}^{-{\frac{yx}{ay-x} \left ({\frac{b \left ( ay-x \right ) ^{2}}{{y}^{2}}}-{\frac{ab \left ( ay-x \right ) }{y}}-d \right ) \left ( -{\frac{ay-x}{y}}+a \right ) ^{-1}}}} \]

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97.5 Problem 5

problem number 827

Added Feb. 17, 2019.

Problem Chapter 4.2.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -a{x}^{3}+{y}^{3} \right ){{\rm e}^{{\frac{cax+by}{a}}}} \]

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97.6 Problem 6

problem number 828

Added Feb. 17, 2019.

Problem Chapter 4.2.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

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

Maple

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

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97.7 Problem 7

problem number 829

Added Feb. 17, 2019.

Problem Chapter 4.2.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica

\[ \text{DSolve}\left [w^{(0,1)}(x,y) \left (a y^2-b x\right )+x (a y+b) w^{(1,0)}(x,y)=a y w(x,y),w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={{\rm e}^{1/9\,\int ^{x}\!{\frac{1}{{\it \_a}\, \left ({\it \_a}\,a-b \right ) } \left ( 2\,{{\rm e}^{\RootOf \left ( -2\,\ln \left ({\frac{ \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ({\it \_a}\,a-b \right ) }{{\it \_a}}} \right ){{\rm e}^{{\it \_Z}}}ax-2\,\ln \left ({\frac{ \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ({\it \_a}\,a-b \right ) }{{\it \_a}}} \right ){{\rm e}^{{\it \_Z}}}ay+2\,\ln \left ( -9\,{\frac{a \left ( y+x \right ) \left ( ax-b \right ) }{x \left ( ay+b \right ) }} \right ){{\rm e}^{{\it \_Z}}}ax+2\,\ln \left ( -9\,{\frac{a \left ( y+x \right ) \left ( ax-b \right ) }{x \left ( ay+b \right ) }} \right ){{\rm e}^{{\it \_Z}}}ay-2\,{{\rm e}^{{\it \_Z}}}\ln \left ( -9/2\,{\frac{ax-b}{ay+b}} \right ) ax-2\,{{\rm e}^{{\it \_Z}}}\ln \left ( -9/2\,{\frac{ax-b}{ay+b}} \right ) ay+2\,{\it \_Z}\,{{\rm e}^{{\it \_Z}}}ax+2\,{\it \_Z}\,{{\rm e}^{{\it \_Z}}}ay+9\,\ln \left ({\frac{ \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ({\it \_a}\,a-b \right ) }{{\it \_a}}} \right ) ax+9\,\ln \left ({\frac{ \left ( 2\,{{\rm e}^{{\it \_Z}}}-9 \right ) \left ({\it \_a}\,a-b \right ) }{{\it \_a}}} \right ) ay-9\,\ln \left ( -9\,{\frac{a \left ( y+x \right ) \left ( ax-b \right ) }{x \left ( ay+b \right ) }} \right ) ax-9\,y\ln \left ( -9\,{\frac{a \left ( y+x \right ) \left ( ax-b \right ) }{x \left ( ay+b \right ) }} \right ) a-2\,y{{\rm e}^{{\it \_Z}}}a+9\,\ln \left ( -9/2\,{\frac{ax-b}{ay+b}} \right ) ax+9\,\ln \left ( -9/2\,{\frac{ax-b}{ay+b}} \right ) ay-9\,{\it \_Z}\,ax-9\,{\it \_Z}\,ay-2\,b{{\rm e}^{{\it \_Z}}}-9\,ax+9\,b \right ) }}b+9\,{\it \_a}\,a-9\,b \right ) }{d{\it \_a}}}}{\it \_F1} \left ( -1/3\,{\frac{1}{a \left ( y+x \right ) } \left ( y\ln \left ( -9\,{\frac{a \left ( y+x \right ) \left ( ax-b \right ) }{x \left ( ay+b \right ) }} \right ) a-\ln \left ( -9/2\,{\frac{ax-b}{ay+b}} \right ) ay+\ln \left ( -9\,{\frac{a \left ( y+x \right ) \left ( ax-b \right ) }{x \left ( ay+b \right ) }} \right ) ax-\ln \left ( -9/2\,{\frac{ax-b}{ay+b}} \right ) ax-ay-b \right ) } \right ) \]

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97.8 Problem 8

problem number 830

Added Feb. 17, 2019.

Problem Chapter 4.2.2.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x(k y-x+a) w_x - y(k x-y +a) w_y = b(y-x) w \]

Mathematica

\[ \text{DSolve}\left [x w^{(1,0)}(x,y) (a+k y-x)-y w^{(0,1)}(x,y) (a+k x-y)=b (y-x) w(x,y),w(x,y),\{x,y\}\right ] \]

Maple

\[ w \left ( x,y \right ) ={\it \_F1} \left ( -1/3\,{\frac{{k}^{2}+k+1}{k \left ( k+1 \right ) } \left ( \left ( k+1 \right ) \ln \left ( -{\frac{ \left ({k}^{2}+k+1 \right ) \left ( a-x-y \right ) k}{ \left ( k-1 \right ) \left ( ky+a-x \right ) }} \right ) +k\ln \left ( x-a \right ) -k\ln \left ( -{\frac{ \left ( k+1 \right ) \left ({k}^{2}+k+1 \right ) \left ( -x+a \right ) }{ \left ( k+2 \right ) \left ( ky+a-x \right ) }} \right ) -\ln \left ( x \right ) -\ln \left ({\frac{ky \left ( k+1 \right ) \left ({k}^{2}+k+1 \right ) }{ \left ( 2\,k+1 \right ) \left ( ky+a-x \right ) }} \right ) \right ) } \right ){{\rm e}^{1/9\,\int ^{x}\!2\,{\frac{b}{ \left ( k+1 \right ) \left ({k}^{2}+k+1 \right ) \left ( a-{\it \_a} \right ) k{\it \_a}} \left ( \left ( k+2 \right ) \left ({\it \_a}\,k-{\it \_a}+a \right ) \left ( k-1 \right ) \left ( k+1/2 \right ) \RootOf \left ({k}^{3}\ln \left ({\it \_a}-a \right ) -{k}^{3}\ln \left ( x-a \right ) -\ln \left ( 2\,{\it \_Z}\,{k}^{2}-{\it \_Z}\,k-3\,{k}^{2}-{\it \_Z}-3\,k-3 \right ){k}^{3}+{k}^{3}\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) +{k}^{2}\ln \left ( x \right ) +{k}^{2}\ln \left ({\it \_a}-a \right ) -{k}^{2}\ln \left ({\it \_a} \right ) -{k}^{2}\ln \left ( x-a \right ) -\ln \left ( 2\,{\it \_Z}\,{k}^{2}-{\it \_Z}\,k-3\,{k}^{2}-{\it \_Z}-3\,k-3 \right ){k}^{2}-{k}^{2}\ln \left ({\it \_Z}\,{k}^{2}+{\it \_Z}\,k+3\,{k}^{2}-2\,{\it \_Z}+3\,k+3 \right ) +2\,{k}^{2}\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) +81\,\int ^{-3\,{\frac{ \left ({k}^{2}+k+1 \right ) \left ( -{k}^{2}y+2\,ka-2\,kx-2\,ky+a-x \right ) }{ \left ( k-1 \right ) \left ( 2\,k+1 \right ) \left ( k+2 \right ) \left ( ky+a-x \right ) }}}\!{\frac{ \left ({k}^{2}+k+1 \right ) ^{3}}{ \left ({\it \_a}\,{k}^{2}+{\it \_a}\,k+3\,{k}^{2}-2\,{\it \_a}+3\,k+3 \right ) \left ( 2\,{\it \_a}\,{k}^{2}-{\it \_a}\,k-3\,{k}^{2}-{\it \_a}-3\,k-3 \right ) \left ( 2\,{\it \_a}\,{k}^{2}+5\,{\it \_a}\,k-3\,{k}^{2}+2\,{\it \_a}-3\,k-3 \right ) }}{d{\it \_a}}{k}^{2}+k\ln \left ( x \right ) +k\ln \left ({\it \_a}-a \right ) -k\ln \left ({\it \_a} \right ) -k\ln \left ( x-a \right ) -\ln \left ( 2\,{\it \_Z}\,{k}^{2}-{\it \_Z}\,k-3\,{k}^{2}-{\it \_Z}-3\,k-3 \right ) k-k\ln \left ({\it \_Z}\,{k}^{2}+{\it \_Z}\,k+3\,{k}^{2}-2\,{\it \_Z}+3\,k+3 \right ) +2\,k\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) +81\,\int ^{-3\,{\frac{ \left ({k}^{2}+k+1 \right ) \left ( -{k}^{2}y+2\,ka-2\,kx-2\,ky+a-x \right ) }{ \left ( k-1 \right ) \left ( 2\,k+1 \right ) \left ( k+2 \right ) \left ( ky+a-x \right ) }}}\!{\frac{ \left ({k}^{2}+k+1 \right ) ^{3}}{ \left ({\it \_a}\,{k}^{2}+{\it \_a}\,k+3\,{k}^{2}-2\,{\it \_a}+3\,k+3 \right ) \left ( 2\,{\it \_a}\,{k}^{2}-{\it \_a}\,k-3\,{k}^{2}-{\it \_a}-3\,k-3 \right ) \left ( 2\,{\it \_a}\,{k}^{2}+5\,{\it \_a}\,k-3\,{k}^{2}+2\,{\it \_a}-3\,k-3 \right ) }}{d{\it \_a}}k+\ln \left ( x \right ) -\ln \left ({\it \_a} \right ) -\ln \left ({\it \_Z}\,{k}^{2}+{\it \_Z}\,k+3\,{k}^{2}-2\,{\it \_Z}+3\,k+3 \right ) +\ln \left ( 2\,{\it \_Z}\,{k}^{2}+5\,{\it \_Z}\,k-3\,{k}^{2}+2\,{\it \_Z}-3\,k-3 \right ) \right ) +3\, \left ( -1/2\,{\it \_a}\,{k}^{2}+ \left ( a-2\,{\it \_a} \right ) k+a/2-{\it \_a}/2 \right ) \left ({k}^{2}+k+1 \right ) \right ) }{d{\it \_a}}}} \]