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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a y -b x}{a}\right ) {\mathrm e}^{\frac {\left (x^{2} a^{2}-3 y^{2} a^{2}+3 x y a b -b^{2} x^{2}\right ) x}{3 a^{3}}}\]

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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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = \left (\frac {x}{y}\right )^{-\frac {c x y}{a y -x}} \mathit {\_F1} \left (\frac {-a y +x}{x y}\right ) {\mathrm e}^{\frac {d y^{2}+\left (a y -x \right ) b x}{a y -x}}\]

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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'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-a x^{3}+y^{3}\right ) {\mathrm e}^{\frac {c a x +b y}{a}}\]

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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'));
 

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

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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 ) = \mathit {\_F1} \left (\frac {a y +\left (x +y \right ) a \ln \left (\frac {-9 a x +9 b}{2 a y +2 b}\right )-\left (x +y \right ) a \ln \left (-\frac {9 \left (x +y \right ) \left (a x -b \right ) a}{\left (a y +b \right ) x}\right )+b}{3 \left (x +y \right ) a}\right ) {\mathrm e}^{\frac {\left (\int _{}^{x}\frac {9 \mathit {\_a} a +2 b \,{\mathrm e}^{\RootOf \left (2 \mathit {\_Z} a x \,{\mathrm e}^{\mathit {\_Z}}+2 \mathit {\_Z} a y \,{\mathrm e}^{\mathit {\_Z}}-2 a x \,{\mathrm e}^{\mathit {\_Z}} \ln \left (\frac {\left (2 \,{\mathrm e}^{\mathit {\_Z}}-9\right ) \left (\mathit {\_a} a -b \right )}{\mathit {\_a}}\right )-2 a x \,{\mathrm e}^{\mathit {\_Z}} \ln \left (-\frac {9 \left (a x -b \right )}{2 \left (a y +b \right )}\right )+2 a x \,{\mathrm e}^{\mathit {\_Z}} \ln \left (-\frac {9 \left (x +y \right ) \left (a x -b \right ) a}{\left (a y +b \right ) x}\right )-2 a y \,{\mathrm e}^{\mathit {\_Z}} \ln \left (\frac {\left (2 \,{\mathrm e}^{\mathit {\_Z}}-9\right ) \left (\mathit {\_a} a -b \right )}{\mathit {\_a}}\right )-2 a y \,{\mathrm e}^{\mathit {\_Z}} \ln \left (-\frac {9 \left (a x -b \right )}{2 \left (a y +b \right )}\right )+2 a y \,{\mathrm e}^{\mathit {\_Z}} \ln \left (-\frac {9 \left (x +y \right ) \left (a x -b \right ) a}{\left (a y +b \right ) x}\right )-9 \mathit {\_Z} a x -9 \mathit {\_Z} a y +9 a x \ln \left (\frac {\left (2 \,{\mathrm e}^{\mathit {\_Z}}-9\right ) \left (\mathit {\_a} a -b \right )}{\mathit {\_a}}\right )+9 a x \ln \left (-\frac {9 \left (a x -b \right )}{2 \left (a y +b \right )}\right )-9 a x \ln \left (-\frac {9 \left (x +y \right ) \left (a x -b \right ) a}{\left (a y +b \right ) x}\right )-2 a y \,{\mathrm e}^{\mathit {\_Z}}+9 a y \ln \left (\frac {\left (2 \,{\mathrm e}^{\mathit {\_Z}}-9\right ) \left (\mathit {\_a} a -b \right )}{\mathit {\_a}}\right )+9 a y \ln \left (-\frac {9 \left (a x -b \right )}{2 \left (a y +b \right )}\right )-9 a y \ln \left (-\frac {9 \left (x +y \right ) \left (a x -b \right ) a}{\left (a y +b \right ) x}\right )-9 a x -2 b \,{\mathrm e}^{\mathit {\_Z}}+9 b \right )}-9 b}{\left (\mathit {\_a} a -b \right ) \mathit {\_a}}d\mathit {\_a} \right )}{9}}\]

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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 ) = \mathit {\_F1} \left (-\frac {\left (k^{2}+k +1\right ) \left (-k \ln \left (-\frac {\left (k +1\right ) \left (k^{2}+k +1\right ) \left (a -x \right )}{\left (k +2\right ) \left (k y +a -x \right )}\right )+k \ln \left (-a +x \right )-\ln \left (x \right )-\ln \left (\frac {\left (k +1\right ) \left (k^{2}+k +1\right ) k y}{\left (2 k +1\right ) \left (k y +a -x \right )}\right )+\left (k +1\right ) \ln \left (-\frac {\left (k^{2}+k +1\right ) \left (a -x -y \right ) k}{\left (k -1\right ) \left (k y +a -x \right )}\right )\right )}{3 \left (k +1\right ) k}\right ) {\mathrm e}^{\frac {\left (\int _{}^{x}\frac {2 \left (\left (\mathit {\_a} k -\mathit {\_a} +a \right ) \left (k +2\right ) \left (k -1\right ) \left (k +\frac {1}{2}\right ) \RootOf \left (k^{3} \ln \left (\mathit {\_a} -a \right )-k^{3} \ln \left (-a +x \right )-k^{3} \ln \left (2 \mathit {\_Z} k^{2}-\mathit {\_Z} k -3 k^{2}-\mathit {\_Z} -3 k -3\right )+k^{3} \ln \left (2 \mathit {\_Z} k^{2}+5 \mathit {\_Z} k -3 k^{2}+2 \mathit {\_Z} -3 k -3\right )+81 k^{2} \left (\int _{}^{-\frac {3 \left (-k^{2} y +2 a k -2 k x -2 k y +a -x \right ) \left (k^{2}+k +1\right )}{\left (k -1\right ) \left (2 k +1\right ) \left (k +2\right ) \left (k y +a -x \right )}}\frac {\left (k^{2}+k +1\right )^{3}}{\left (\mathit {\_a} k^{2}+\mathit {\_a} k +3 k^{2}-2 \mathit {\_a} +3 k +3\right ) \left (2 \mathit {\_a} k^{2}-\mathit {\_a} k -3 k^{2}-\mathit {\_a} -3 k -3\right ) \left (2 \mathit {\_a} k^{2}+5 \mathit {\_a} k -3 k^{2}+2 \mathit {\_a} -3 k -3\right )}d\mathit {\_a} \right )-k^{2} \ln \left (\mathit {\_a} \right )+k^{2} \ln \left (x \right )+k^{2} \ln \left (\mathit {\_a} -a \right )-k^{2} \ln \left (-a +x \right )-k^{2} \ln \left (\mathit {\_Z} k^{2}+\mathit {\_Z} k +3 k^{2}-2 \mathit {\_Z} +3 k +3\right )-k^{2} \ln \left (2 \mathit {\_Z} k^{2}-\mathit {\_Z} k -3 k^{2}-\mathit {\_Z} -3 k -3\right )+2 k^{2} \ln \left (2 \mathit {\_Z} k^{2}+5 \mathit {\_Z} k -3 k^{2}+2 \mathit {\_Z} -3 k -3\right )+81 k \left (\int _{}^{-\frac {3 \left (-k^{2} y +2 a k -2 k x -2 k y +a -x \right ) \left (k^{2}+k +1\right )}{\left (k -1\right ) \left (2 k +1\right ) \left (k +2\right ) \left (k y +a -x \right )}}\frac {\left (k^{2}+k +1\right )^{3}}{\left (\mathit {\_a} k^{2}+\mathit {\_a} k +3 k^{2}-2 \mathit {\_a} +3 k +3\right ) \left (2 \mathit {\_a} k^{2}-\mathit {\_a} k -3 k^{2}-\mathit {\_a} -3 k -3\right ) \left (2 \mathit {\_a} k^{2}+5 \mathit {\_a} k -3 k^{2}+2 \mathit {\_a} -3 k -3\right )}d\mathit {\_a} \right )-k \ln \left (\mathit {\_a} \right )+k \ln \left (x \right )+k \ln \left (\mathit {\_a} -a \right )-k \ln \left (-a +x \right )-k \ln \left (\mathit {\_Z} k^{2}+\mathit {\_Z} k +3 k^{2}-2 \mathit {\_Z} +3 k +3\right )-k \ln \left (2 \mathit {\_Z} k^{2}-\mathit {\_Z} k -3 k^{2}-\mathit {\_Z} -3 k -3\right )+2 k \ln \left (2 \mathit {\_Z} k^{2}+5 \mathit {\_Z} k -3 k^{2}+2 \mathit {\_Z} -3 k -3\right )-\ln \left (\mathit {\_a} \right )+\ln \left (x \right )-\ln \left (\mathit {\_Z} k^{2}+\mathit {\_Z} k +3 k^{2}-2 \mathit {\_Z} +3 k +3\right )+\ln \left (2 \mathit {\_Z} k^{2}+5 \mathit {\_Z} k -3 k^{2}+2 \mathit {\_Z} -3 k -3\right )\right )+3 \left (k^{2}+k +1\right ) \left (-\frac {\mathit {\_a} k^{2}}{2}-\frac {\mathit {\_a}}{2}+\frac {a}{2}+\left (-2 \mathit {\_a} +a \right ) k \right )\right ) b}{\left (k +1\right ) \left (k^{2}+k +1\right ) \left (-\mathit {\_a} +a \right ) \mathit {\_a} k}d\mathit {\_a} \right )}{9}}\]

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