6.2.18 6.4

6.2.18.1 [681] problem number 1

problem number 681

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'));
 

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

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6.2.18.2 [682] problem number 2

problem number 682

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'));
 

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

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6.2.18.3 [683] problem number 3

problem number 683

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'));
 

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

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6.2.18.4 [684] problem number 4

problem number 684

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'));
 

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

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6.2.18.5 [685] problem number 5

problem number 685

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( \left ( 4\,\LegendreP \left ( 1/2\,{\frac {2\,a+3\,\lambda }{\lambda }},1/2\,{\frac {-\lambda +2\,a}{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\lambda -2\, \left ( 1/2\,\lambda \, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}+y \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sin \left ( \lambda \,x \right ) + \left ( \left ( a+3/2\,\lambda \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-\lambda /2 \right ) \cos \left ( \lambda \,x \right ) -y\sin \left ( \lambda \,x \right ) \right ) \LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {-\lambda +2\,a}{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \right ) \left ( -4\,\LegendreQ \left ( 1/2\,{\frac {2\,a+3\,\lambda }{\lambda }},1/2\,{\frac {-\lambda +2\,a}{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\lambda +2\, \left ( 1/2\,\lambda \, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}+y \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sin \left ( \lambda \,x \right ) + \left ( \left ( a+3/2\,\lambda \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-\lambda /2 \right ) \cos \left ( \lambda \,x \right ) -y\sin \left ( \lambda \,x \right ) \right ) \LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {-\lambda +2\,a}{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \right ) ^{-1} \right ) \]

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6.2.18.6 [686] problem number 6

problem number 686

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -1/2\,{\frac {{{\rm e}^{-2\,ibx}} \left ( i\cot \left ( ax \right ) a-iy-b \right ) }{b \left ( ib-a\cot \left ( ax \right ) +y \right ) }} \right ) \]

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6.2.18.7 [687] problem number 7

problem number 687

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'));
 

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

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6.2.18.8 [688] problem number 8

problem number 688

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'));
 

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

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6.2.18.9 [689] problem number 9

problem number 689

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'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a\mu }\ln \left ( {{\rm e}^{-{\frac {a\mu \,\pi }{\lambda }}}}\sqrt {{\frac {1}{ \left ( \cos \left ( \mu \,y \right ) \right ) ^{2}}{{\rm e}^{2\,{\frac {\mu \, \left ( \left ( \lambda \,x+\pi /2 \right ) \sin \left ( \lambda \,x \right ) +\cos \left ( \lambda \,x \right ) \right ) a}{\lambda \,\sin \left ( \lambda \,x \right ) }}}}}}\sin \left ( \mu \,y \right ) \cos \left ( \mu \,y \right ) \right ) } \right ) \]

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6.2.18.10 [690] problem number 10

problem number 690

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'));
 

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

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6.2.18.11 [691] problem number 11

problem number 691

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'));
 

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

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6.2.18.12 [692] problem number 12

problem number 692

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'));
 

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

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