6.4.3 2.2

   6.4.3.1 [1031] Problem 1
   6.4.3.2 [1032] Problem 2
   6.4.3.3 [1033] Problem 3
   6.4.3.4 [1034] Problem 4
   6.4.3.5 [1035] Problem 5
   6.4.3.6 [1036] Problem 6
   6.4.3.7 [1037] Problem 7
   6.4.3.8 [1038] Problem 8

6.4.3.1 [1031] Problem 1

problem number 1031

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

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

Maple

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

________________________________________________________________________________________

6.4.3.2 [1032] Problem 2

problem number 1032

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

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

Maple

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

________________________________________________________________________________________

6.4.3.3 [1033] Problem 3

problem number 1033

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

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

Maple

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

________________________________________________________________________________________

6.4.3.4 [1034] Problem 4

problem number 1034

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

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

Maple

\[w \left ( x,y \right ) ={\it \_F1} \left ({\frac{-ya+x}{xy}} \right ) \left ({\frac{x}{y}} \right ) ^{-{\frac{cxy}{ya-x}}}{{\rm e}^{{\frac{x \left ( ya-x \right ) b+d{y}^{2}}{ya-x}}}}\]

________________________________________________________________________________________

6.4.3.5 [1035] Problem 5

problem number 1035

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

\begin{align*} & \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 \}\\ \end{align*}

Maple

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

________________________________________________________________________________________

6.4.3.6 [1036] Problem 6

problem number 1036

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

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

Maple

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

________________________________________________________________________________________

6.4.3.7 [1037] Problem 7

problem number 1037

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

$Aborted

Maple

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

________________________________________________________________________________________

6.4.3.8 [1038] Problem 8

problem number 1038

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

Failed

Maple

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

________________________________________________________________________________________