6.2.17 6.3

6.2.17.1 [665] problem number 1

problem number 665

Added January 14, 2019.

Problem 2.6.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-a x-b x+\frac {\log (\cos (\lambda x))}{\lambda }+y\right )\right \}\right \}\]

Maple ✓

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

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

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6.2.17.2 [666] problem number 2

problem number 666

Added January 14, 2019.

Problem 2.6.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-x+\frac {-i (a+b-i) \log (-\tan (\lambda y)+i)+i (a+b+i) \log (\tan (\lambda y)+i)+2 \log (a+b+\tan (\lambda y))}{2 \lambda (a+b-i) (a+b+i)}\right )\right \}\right \}\]

Maple ✓

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

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

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6.2.17.3 [667] problem number 3

problem number 667

Added January 14, 2019.

Problem 2.6.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

Has unresolved integrals

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6.2.17.4 [668] problem number 4

problem number 668

Added January 14, 2019.

Problem 2.6.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( y^2+ a *lambda + a*(lambda -a) *tan(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'));
 

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

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6.2.17.5 [669] problem number 5

problem number 669

Added January 14, 2019.

Problem 2.6.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(  y^2+ lambda^2 +3*a*lambda +a*(lambda-a)*tan(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'));
 

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

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6.2.17.6 [670] problem number 6

problem number 670

Added January 14, 2019.

Problem 2.6.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\exp \left (-\int _1^x-a K[1] \tan ^k(b K[1])dK[1]\right )}{x^2 y+x}-\int _1^x\frac {\exp \left (-\int _1^{K[2]}-a K[1] \tan ^k(b K[1])dK[1]\right )}{K[2]^2}dK[2]\right )\right \}\right \}\]

Maple ✓

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

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

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6.2.17.7 [671] problem number 7

problem number 671

Added January 14, 2019.

Problem 2.6.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x -\left ( (k+1) x^k y^2- a x^{k+1} (\tan x)^m y + a(\tan x)^m \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - a*x^(k + 1)*Tan[x]^m*y + a*Tan[x]^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {(k+1) \left (y x^{k+1}-1\right )}{(k+1) \left (y x^{k+1}-1\right ) \int _1^x\exp \left (\int _1^{K[2]}-\frac {-a \tan ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )dK[2]-x \exp \left (\int _1^x-\frac {-a \tan ^m(K[1]) K[1]^{k+2}+k+2}{K[1]}dK[1]\right )}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)-(  (k+1)*x^k*y^2- a*x^(k+1)*tan(x)^m*y + a*tan(x)^m )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-\left (x^{k +1} y -1\right ) \left (k +1\right ) \int x^{k} {\mathrm e}^{\int \frac {\tan \left (x \right )^{m} x^{k +1} a x -2 k -2}{x}d x}d x +x^{k +1} {\mathrm e}^{\int \frac {\tan \left (x \right )^{m} x^{k +1} a x -2 k -2}{x}d x}}{x^{k +1} y -1}\right )\]

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6.2.17.8 [672] problem number 8

problem number 672

Added January 20, 2019.

Problem 2.6.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +\left ( a \tan ^n(\lambda x) y^2- a b^2 \tan ^{n+2}(\lambda x) + b \lambda \tan ^2(\lambda x)+ b \lambda \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Tan[lambda*x]^n*y^2 - a*b^2*Tan[lambda*x]^(n + 2) + b*lambda*Tan[lambda*x]^2 + b*lambda)*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)+(a*tan(lambda*x)^n*y^2- a*b^2*tan(lambda*x)^(n+2) + b*lambda*tan(lambda*x)^2+ b*lambda)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

time expired

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6.2.17.9 [673] problem number 9

problem number 673

Added January 20, 2019.

Problem 2.6.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x +\left ( a \tan ^k(\lambda x+\mu )(y-b x^n-c)^2 + y- b x^n + b n x^{n-1}-c \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (a*Tan[lambda*x + mu]^k*(y - b*x^n - c)^2 + y - b*x^n + b*n*x^(n - 1) - c)*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)+(a *tan(lambda*x+mu)^k*(y-b*x^n-c)^2 + y- b*x^n + b*n*x^(n-1)-c)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {\left (-y +b \,x^{n}+c \right ) a \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k} \left (\tan \left (\mu \right ) \tan \left (\lambda x \right )-1\right )^{k}}{\left (\left (-y +b \,x^{n}+c \right ) a \left (\frac {-\tan \left (\mu \right )-\tan \left (\lambda x \right )}{\tan \left (\mu \right ) \tan \left (\lambda x \right )-1}\right )^{k} \left (\tan \left (\mu \right ) \tan \left (\lambda x \right )-1\right )^{k} \int \left (\tan \left (\mu \right ) \tan \left (\lambda x \right )-1\right )^{-k} \left (\tan \left (\mu \right )+\tan \left (\lambda x \right )\right )^{k} \left (\cos \left (\lambda x \right )-i \sin \left (\lambda x \right )\right )^{\frac {i}{\lambda }}d x -\left (\tan \left (\mu \right )+\tan \left (\lambda x \right )\right )^{k} \left (\cos \left (\lambda x \right )-i \sin \left (\lambda x \right )\right )^{\frac {i}{\lambda }}\right ) \lambda }\right )\]

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6.2.17.10 [674] problem number 10

problem number 674

Added January 20, 2019.

Problem 2.6.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x +\left ( a \tan ^m(\lambda x)y^2 +k y+ a b^2 x^{2 k} \tan ^m(\lambda x) \right ) w_y = 0 \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^xa K[1]^{k-1} \tan ^m(\lambda K[1])dK[1]\right )\right \}\right \}\]

Maple ✓

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

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

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6.2.17.11 [675] problem number 11

problem number 675

Added January 20, 2019.

Problem 2.6.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a \tan (\lambda x)+b) w_x +\left ( y^2+ c \tan (\mu x) y - k^2 + c k \tan (\mu x) \right ) w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

restart; 
pde :=   (a*tan(lambda*x)+b)*diff(w(x,y),x)+ (y^2+ c *tan(mu*x)*y - k^2 + c*k*tan(mu*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 {{\mathrm e}^{\frac {2 a \lambda \left (i a -b \right ) c \left (a^{2}+b^{2}\right ) \int \frac {{\mathrm e}^{2 i \mu x}-1}{\left (\left (i b +a \right ) {\mathrm e}^{2 i \lambda x}+i b -a \right ) \left ({\mathrm e}^{2 i \mu x}+1\right ) \left (i a -b \right )}d x -2 k b \arctan \left (\tan \left (\lambda x \right )\right ) \left (i b +a \right )-c x \lambda \left (a^{2}+b^{2}\right )}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \lambda }} \left (\int \left (\sec \left (\lambda x \right )^{2}\right )^{\frac {k a}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{\frac {2 a \lambda \left (i a -b \right ) c \left (a^{2}+b^{2}\right ) \int \frac {{\mathrm e}^{2 i \mu x}-1}{\left (\left (i b +a \right ) {\mathrm e}^{2 i \lambda x}+i b -a \right ) \left ({\mathrm e}^{2 i \mu x}+1\right ) \left (i a -b \right )}d x -2 k b \arctan \left (\tan \left (\lambda x \right )\right ) \left (i b +a \right )-c x \lambda \left (a^{2}+b^{2}\right )}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \lambda }} \left ({\mathrm e}^{2 i \mu x}+1\right )^{\frac {c}{\mu \left (i a -b \right )}} \left (a \tan \left (\lambda x \right )+b \right )^{\frac {\left (-a^{2}-b^{2}\right ) \lambda -2 a k}{\lambda \left (a^{2}+b^{2}\right )}}d x \left (k +y \right ) {\mathrm e}^{\frac {-2 a \lambda \left (i a -b \right ) c \left (a^{2}+b^{2}\right ) \int \frac {{\mathrm e}^{2 i \mu x}-1}{\left (\left (i b +a \right ) {\mathrm e}^{2 i \lambda x}+i b -a \right ) \left ({\mathrm e}^{2 i \mu x}+1\right ) \left (i a -b \right )}d x +2 k b \arctan \left (\tan \left (\lambda x \right )\right ) \left (i b +a \right )+c x \lambda \left (a^{2}+b^{2}\right )}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \lambda }}+\left ({\mathrm e}^{2 i \mu x}+1\right )^{\frac {c}{\mu \left (i a -b \right )}} \left (\sec \left (\lambda x \right )^{2}\right )^{\frac {k a}{\lambda \left (a^{2}+b^{2}\right )}} \left (a \tan \left (\lambda x \right )+b \right )^{-\frac {2 k a}{\lambda \left (a^{2}+b^{2}\right )}}\right )}{k +y}\right )\]

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6.2.17.12 [676] problem number 12

problem number 676

Added January 20, 2019.

Problem 2.6.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n y^m + b x) w_x + \tan ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*y^m + b*x)*D[w[x, y], x] + Tan[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 :=   (a*x^n*y^m + b*x)*diff(w(x,y),x)+ tan(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'));
 

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

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6.2.17.13 [677] problem number 13

problem number 677

Added January 20, 2019.

Problem 2.6.3.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n + b x \tan ^m y) w_x + y^k w_y = 0 \]

Mathematica ✗

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

Failed

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (a \left (n -1\right ) \int y^{-k} {\mathrm e}^{b \int \tan \left (y \right )^{m} y^{-k}d y \left (n -1\right )}d y +x^{-n +1} {\mathrm e}^{b \int \tan \left (y \right )^{m} y^{-k}d y \left (n -1\right )}\right )\]

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6.2.17.14 [678] problem number 14

problem number 678

Added January 20, 2019.

Problem 2.6.3.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n + b x \tan ^m y) w_x + \tan ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n + b*x*Tan[y]^m)*D[w[x, y], x] + Tan[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 :=   (a*x^n + b*x*tan(y)^m)*diff(w(x,y),x)+  tan(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'));
 

time expired

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6.2.17.15 [679] problem number 15

problem number 679

Added January 20, 2019.

Problem 2.6.3.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ (a x^n \tan ^m y + b x) w_x + \tan ^k(\lambda y) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  (a*x^n*Tan[y]^m + b*x)*D[w[x, y], x] + Tan[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 :=   (a*x^n*tan(y)^m+ b*x)*diff(w(x,y),x)+  tan(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'));
 

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

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