6.2.18 6.4

6.2.18.1 [680] problem number 1

problem number 680

Added January 20, 2019.

Problem 2.6.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \cot ^k(\lambda x)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Cot[lambda*x]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a \cot ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }-b x+y\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   diff(w(x,y),x)+  (a*cot(lambda*x)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-b x +y -\left (\int a \left (\cot ^{k}\left (\lambda x \right )\right )d x \right )\right )\] Has unresolved integral

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6.2.18.2 [681] problem number 2

problem number 681

Added January 20, 2019.

Problem 2.6.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \cot ^k(\lambda y)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Cot[lambda*y]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{a \cot ^k(\lambda K[1])+b}dK[1]-x\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   diff(w(x,y),x)+  (a*cot(lambda*y)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (x -\left (\int \frac {1}{a \left (\cot ^{k}\left (\lambda y \right )\right )+b}d y \right )\right )\]

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6.2.18.3 [682] problem number 3

problem number 682

Added January 20, 2019.

Problem 2.6.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \cot ^k(x+\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Cot[x + lambda*y]^k*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=   diff(w(x,y),x)+  cot(x+lambda*y)^k*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\lambda \left (\int _{}^{\frac {\lambda y +x}{\lambda }}\frac {1}{\lambda \left (\cot ^{k}\left (\mathit {\_a} \lambda \right )\right )+1}d\mathit {\_a} \right )+x \right )\]

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6.2.18.4 [683] problem number 4

problem number 683

Added January 20, 2019.

Problem 2.6.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+a \lambda + a(\lambda -a) \cot ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + a*lambda + a*(lambda - a)*Cot[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=   diff(w(x,y),x)+ ( y^2+a*lambda + a*(lambda-a)*cot(lambda*x)^2  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {-2 \left (\cos \left (2 \lambda x \right )-1\right ) \lambda \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )+\left (-a \cos \left (\lambda x \right )+a \cos \left (3 \lambda x \right )+3 y \sin \left (\lambda x \right )-y \sin \left (3 \lambda x \right )\right ) \LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )}{2 \left (\cos \left (2 \lambda x \right )-1\right ) \lambda \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )+\left (a \cos \left (\lambda x \right )-a \cos \left (3 \lambda x \right )-3 y \sin \left (\lambda x \right )+y \sin \left (3 \lambda x \right )\right ) \LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )}\right )\]

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6.2.18.5 [684] problem number 5

problem number 684

Added January 20, 2019.

Problem 2.6.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+\lambda ^2 + 3 a \lambda +a(\lambda -a) \cot ^2(\lambda x) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda^2 + 3*a*lambda + a*(lambda - a)*Cot[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=   diff(w(x,y),x)+ (  y^2+lambda^2 + 3*a*lambda +a*(lambda-a)*cot(lambda*x)^2   )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {4 \lambda \LegendreP \left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \left (\sin ^{2}\left (\lambda x \right )\right )-2 \left (\frac {\lambda \left (\cos ^{3}\left (\lambda x \right )\right )}{2}+y \left (\cos ^{2}\left (\lambda x \right )\right ) \sin \left (\lambda x \right )-y \sin \left (\lambda x \right )+\left (\left (a +\frac {3 \lambda }{2}\right ) \left (\sin ^{2}\left (\lambda x \right )\right )-\frac {\lambda }{2}\right ) \cos \left (\lambda x \right )\right ) \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )}{-4 \lambda \LegendreQ \left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \left (\sin ^{2}\left (\lambda x \right )\right )+2 \left (\frac {\lambda \left (\cos ^{3}\left (\lambda x \right )\right )}{2}+y \left (\cos ^{2}\left (\lambda x \right )\right ) \sin \left (\lambda x \right )-y \sin \left (\lambda x \right )+\left (\left (a +\frac {3 \lambda }{2}\right ) \left (\sin ^{2}\left (\lambda x \right )\right )-\frac {\lambda }{2}\right ) \cos \left (\lambda x \right )\right ) \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )}\right )\]

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6.2.18.6 [685] problem number 6

problem number 685

Added January 20, 2019.

Problem 2.6.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2-2 a \cot (a x) y + b^2-a^2 \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - 2*a*Cot[a*x]*y + b^2 - a^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=   diff(w(x,y),x)+ (  y^2-2*a*cot(a*x)*y + b^2-a^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.2.18.7 [686] problem number 7

problem number 686

Added January 20, 2019.

Problem 2.6.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\lambda x) w_x + a \cot (\mu y) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  Cot[lambda*x]*D[w[x, y], x] + a*Cot[mu*y]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=   cot(lambda*x)*diff(w(x,y),x)+ a*cot(mu*y)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {a \mu \ln \left (\frac {\cot ^{2}\left (\lambda x \right )+1}{\cot \left (\lambda x \right )^{2}}\right )+\lambda \ln \left (\cos ^{2}\left (\mu y \right )\right )}{2 \lambda \mu }\right )\]

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6.2.18.8 [687] problem number 8

problem number 687

Added January 20, 2019.

Problem 2.6.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\mu y) w_x + a \cot (\lambda x) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  Cot[mu*y]*D[w[x, y], x] + a*Cot[lambda*x]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=   cot(mu*y)*diff(w(x,y),x)+ a*cot(lambda*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\ln \left (\frac {\sqrt {\left (\tan ^{2}\left (\mu y \right )+1\right ) \left (-\frac {2}{\cos \left (2 \lambda x \right )-1}\right )^{\frac {a \mu }{\lambda }}}\, \tan \left (\mu y \right )}{\tan ^{2}\left (\mu y \right )+1}\right )}{a \mu }\right )\]

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6.2.18.9 [688] problem number 9

problem number 688

Added January 20, 2019.

Problem 2.6.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\mu y) w_x + a \cot ^2(\lambda x) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  Cot[mu*y]*D[w[x, y], x] + a*Cot[lambda*x]^2*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {4 \sin (\mu y) e^{\frac {a \mu (\lambda x+\cot (\lambda x))}{\lambda }}}{\mu }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   cot(mu*y)*diff(w(x,y),x)+ a*cot(lambda*x)^2*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\lambda \ln \left (\frac {\tan ^{2}\left (\mu y \right )}{\tan ^{2}\left (\mu y \right )+1}\right ) \sin \left (\lambda x \right )+2 \left (\mathrm {arccot}\left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right ) \sin \left (\lambda x \right )+\cos \left (\lambda x \right )-\frac {\pi \sin \left (\lambda x \right )}{2}\right ) a \mu }{2 a \lambda \mu \sin \left (\lambda x \right )}\right )\]

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6.2.18.10 [689] problem number 10

problem number 689

Added January 20, 2019.

Problem 2.6.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (y+a) w_x + c \cot (x+b) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  Cot[y + a]*D[w[x, y], x] + c*Cot[x + b]^2*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=   cot(y+a)*diff(w(x,y),x)+ c*cot(x+b)^2*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (\tan \left (b \right )+\tan \left (x \right )\right ) \ln \left (\frac {1}{\sin \left (y \right )^{2}}\right ) \tan \left (b \right )-2 \left (\tan \left (b \right )+\tan \left (x \right )\right ) \ln \left (\frac {\cos \left (y \right ) \tan \left (a \right )+\sin \left (y \right )}{\sin \left (y \right ) \tan \left (a \right )}\right ) \tan \left (b \right )+2 \left (\left (-x +\tan \left (x \right )+\frac {\pi }{2}\right ) \left (\tan ^{2}b \right )-\left (x -\frac {\pi }{2}\right ) \tan \left (b \right ) \tan \left (x \right )+\tan \left (x \right )\right ) c}{2 \left (\tan \left (b \right )+\tan \left (x \right )\right ) \tan \left (b \right )}\right )\]

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6.2.18.11 [690] problem number 11

problem number 690

Added January 20, 2019.

Problem 2.6.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \cot (\lambda x) \cot (\mu y) w_x + a w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  Cot[lambda*x]*Cot[mu*y]*D[w[x, y], x] + a*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \sin (\mu y) \cos ^{\frac {a \mu }{\lambda }}(\lambda x)}{\mu }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=   cot(lambda*x)*cot(mu*y)*diff(w(x,y),x)+ a*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\ln \left (\frac {\sqrt {\left (\tan ^{2}\left (\mu y \right )+1\right ) \left (\cos ^{\frac {2 a \mu }{\lambda }}\left (\lambda x \right )\right )}\, \tan \left (\mu y \right )}{\tan ^{2}\left (\mu y \right )+1}\right )}{a \mu }\right )\]

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6.2.18.12 [691] problem number 12

problem number 691

Added January 20, 2019.

Problem 2.6.4.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ \cot (\lambda x) \cot (\mu y) w_x + a \cot (v x) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  Cot[lambda*x]*Cot[mu*y]*D[w[x, y], x] + a*Cot[v*x]*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=   cot(lambda*x)*cot(mu*y)*diff(w(x,y),x)+ a*cot(v*x)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\ln \left (\left ({\mathrm e}^{2 i v x}-1\right )^{\frac {i a \mu }{v}} \mathrm {csgn}\left (\frac {1}{\cos \left (\mu y \right )}\right ) {\mathrm e}^{a \mu x} {\mathrm e}^{\int -\frac {\left (-2 \,{\mathrm e}^{2 i v x}-2\right ) a \mu }{\left ({\mathrm e}^{2 i v x}-1\right ) \left ({\mathrm e}^{2 i \lambda x}+1\right )}d x} \sin \left (\mu y \right )\right )}{a \mu }\right )\]

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