6.4.3 2.2

6.4.3.1 [1030] Problem 1

problem number 1030

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 ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (x^2 - y^2)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x) +b*diff(w(x,y),y) = (x^2-y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.4.3.2 [1031] Problem 2

problem number 1031

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 ✓

ClearAll["Global`*"]; 
pde =  x^2*D[w[x, y], x] + a*x*y*D[w[x, y], y] == b*y^2*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  x^2*diff(w(x,y),x) +a*x*y*diff(w(x,y),y) = b*y^2*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.4.3.3 [1032] Problem 3

problem number 1032

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 ✓

ClearAll["Global`*"]; 
pde =  a*x^2*D[w[x, y], x] + b*y^2*D[w[x, y], y] == (x + c*y)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde := a*x^2*diff(w(x,y),x) +b*y^2*diff(w(x,y),y) = (x+c*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.4.3.4 [1033] Problem 4

problem number 1033

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 ✓

ClearAll["Global`*"]; 
pde =  x^2*D[w[x, y], x] + a*y^2*D[w[x, y], y] == (b*x^2 + c*x*y + d*y^2)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=x^2*diff(w(x,y),x) +a*y^2*diff(w(x,y),y) = (b*x^2+c*x*y+d*y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.4.3.5 [1034] Problem 5

problem number 1034

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 ✓

ClearAll["Global`*"]; 
pde =  y^2*D[w[x, y], x] + a*x^2*D[w[x, y], y] == (b*x^2 + c*y^2)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\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 ✓

restart; 
pde :=y^2*diff(w(x,y),x) +a*x^2*diff(w(x,y),y) =(b*x^2+c*y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.4.3.6 [1035] Problem 6

problem number 1035

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 ✓

ClearAll["Global`*"]; 
pde =  x*y*D[w[x, y], x] + a*y^2*D[w[x, y], y] == (b*x + c*y + d)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=x*y*diff(w(x,y),x) +a*y^2*diff(w(x,y),y) =(b*x+c*y+d)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.4.3.7 [1036] Problem 7

problem number 1036

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 ✗

ClearAll["Global`*"]; 
pde =  x*(a*y + b)*D[w[x, y], x] + (a*y^2 - b*x)*D[w[x, y], y] == a*y*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=x*(a*y+b)*diff(w(x,y),x) +(a*y^2-b*x)*diff(w(x,y),y) =a*y*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.4.3.8 [1037] Problem 8

problem number 1037

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 ✗

ClearAll["Global`*"]; 
pde =  x*(k*y - x + a)*D[w[x, y], x] - y*(k*x - y + a)*D[w[x, y], y] == b*(y - x)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=x*(k*y-x+a)*diff(w(x,y),x)-y*(k*x-y+a)*diff(w(x,y),y) = b*(y-x)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime')); 
sol:=simplify(sol);
 

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

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