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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {a \cot ^{k+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {k+1}{2},\frac {k+3}{2},-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }-b x+y\right )\right \}\right \}\]

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'));
 

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

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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {\lambda c_1}{a \lambda \cot ^k(x+\lambda K[6607])+1}-\int _1^x-\frac {a k \lambda ^2 c_1 \cot ^k(K[1]+\lambda K[6607]) \csc (K[1]+\lambda K[6607]) \sec (K[1]+\lambda K[6607])}{\left (a \lambda \cot ^k(K[1]+\lambda K[6607])+1\right )^2}dK[1]\right )dK[6607]+\int _1^x\frac {a \lambda c_1 \cot ^k(\lambda y+K[1])}{a \lambda \cot ^k(\lambda y+K[1])+1}dK[1]+c_2\right \}\right \}\]

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'));
 

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

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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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\sin ^{\frac {a}{\lambda }}(2 \lambda x) e^{\text {arctanh}(\cos (2 \lambda x))} ((a+\lambda ) \cos (2 \lambda x)+a+y \sin (2 \lambda x)-\lambda )}{\sin ^{\frac {a}{\lambda }}(2 \lambda x) e^{\text {arctanh}(\cos (2 \lambda x))} ((a+\lambda ) \cos (2 \lambda x)+a+y \sin (2 \lambda x)-\lambda ) \int _1^xe^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda K[1]))}{\lambda }} \sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda K[1])dK[1]+e^{\frac {a \text {arctanh}(\cos (2 \lambda x))}{\lambda }}}\right )\right \}\right \}\]

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'));
 

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

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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]];
 

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

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {{\mathrm e}^{-2 i b x} \left (-b +i a \cot \left (a x \right )-i y \right )}{2 b \left (-a \cot \left (a x \right )+y +i b \right )}\right )\]

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\frac {a \mu \left (\ln \left (2\right )+\ln \left (-\frac {1}{-1+\cos \left (2 \lambda x \right )}\right )\right )}{2}+\ln \left (\operatorname {csgn}\left (\sec \left (\mu y \right )\right ) \sin \left (\mu y \right )\right ) \lambda }{\lambda a \mu }\right )\]

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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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\ln \left (\operatorname {csgn}\left (\sec \left (\mu y \right )\right ) \sin \left (\mu y \right )\right ) \lambda +\left (\operatorname {arccot}\left (\cot \left (\lambda x \right )\right )-\frac {\pi }{2}+\cot \left (\lambda x \right )\right ) \mu a}{a \mu \lambda }\right )\]

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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'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {\left (\tan \left (x \right )+\tan \left (b \right )\right ) \ln \left (\csc \left (y \right )^{2}\right )+\left (-2 \tan \left (b \right )-2 \tan \left (x \right )\right ) \ln \left (\cot \left (a \right )+\cot \left (y \right )\right )+2 c \left (\left (\tan \left (b \right )-x +\frac {\pi }{2}+\cot \left (b \right )\right ) \tan \left (x \right )-\tan \left (b \right ) \left (x -\frac {\pi }{2}\right )\right )}{2 \tan \left (x \right )+2 \tan \left (b \right )}\right )\]

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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'));
 

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

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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]];
 

\[\left \{\left \{w(x,y)\to -\frac {c_1 \log (\sin (\mu y))}{4 a \mu }+\frac {1}{4} c_1 \int \tan (\lambda x) \cot (v x) \, dx+c_2\right \}\right \}\]

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'));
 

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

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