6.5.17 6.4

6.5.17.1 [1309] Problem 1

problem number 1309

Added April 11, 2019.

Problem Chapter 5.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + k \cot (\lambda x+\mu y) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {k \cot \left (\frac {-\left (-\mathit {\_a} +x \right ) b \mu +\left (\mathit {\_a} \lambda +\mu y \right ) a}{a}\right ) {\mathrm e}^{-\frac {\mathit {\_a} c}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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6.5.17.2 [1310] Problem 2

problem number 1310

Added April 11, 2019.

Problem Chapter 5.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = w + c_1 \cot ^k(\lambda x) + c_2 \cot ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*Cot[lambda*x]^k + c2*Cot[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \cot ^k(\lambda K[1])+\text {c2} \cot ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+ c1*cot(lambda*x)^k + c2*cot(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \left (\cot ^{k}\left (\mathit {\_a} \lambda \right )\right )+\mathit {c2} \left (\cot ^{n}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )\right )\right ) {\mathrm e}^{-\frac {\mathit {\_a}}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]

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6.5.17.3 [1311] Problem 3

problem number 1311

Added April 11, 2019.

Problem Chapter 5.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + \cot ^k(\lambda x) \cot ^n(\beta y) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\cot ^{k}\left (\mathit {\_a} \lambda \right )\right ) \left (\cot ^{n}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {\mathit {\_a} c}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

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6.5.17.4 [1312] Problem 4

problem number 1312

Added April 11, 2019.

Problem Chapter 5.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cot (\mu y) w_y = c \cot (\lambda x) w + k \cot (\nu x) \]

Mathematica ✓

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

\begin {align*} & \left \{w(x,y)\to \sin ^{\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x\frac {k \cot (\nu K[1]) \sin ^{-\frac {c}{a \lambda }}(\lambda K[1])}{a}dK[1]+c_1\left (\frac {\log (\sec (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\\& \left \{w(x,y)\to \sin ^{\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x\frac {k \cot (\nu K[2]) \sin ^{-\frac {c}{a \lambda }}(\lambda K[2])}{a}dK[2]+c_1\left (\frac {\log (\sec (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\\ \end {align*}

Maple ✓

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

\[w \left (x , y\right ) = \left (-\left (\int _{}^{y}\frac {\left (\sin \left (\frac {2 a \nu \ln \left (\frac {\cos \left (\mathit {\_a} \mu \right )}{\sin \left (\mathit {\_a} \mu \right )}\right )-a \nu \ln \left (-\frac {2}{\cos \left (2 \mathit {\_a} \mu \right )-1}\right )+a \nu \ln \left (\cot ^{2}\left (\mu y \right )+1\right )-2 a \nu \ln \left (\cot \left (\mu y \right )\right )-2 \left (-\mathit {\_a} \mu +\nu x \right ) b \mu }{2 b \mu }\right )-\sin \left (\frac {2 a \nu \ln \left (\frac {\cos \left (\mathit {\_a} \mu \right )}{\sin \left (\mathit {\_a} \mu \right )}\right )-a \nu \ln \left (-\frac {2}{\cos \left (2 \mathit {\_a} \mu \right )-1}\right )+a \nu \ln \left (\cot ^{2}\left (\mu y \right )+1\right )-2 a \nu \ln \left (\cot \left (\mu y \right )\right )-2 \left (\mathit {\_a} \mu +\nu x \right ) b \mu }{2 b \mu }\right )\right ) k \left (-\sin \left (\frac {\left (b \mu x -a \ln \left (\frac {\cos \left (\mathit {\_a} \mu \right )}{\sin \left (\mathit {\_a} \mu \right )}\right )+\frac {a \ln \left (-\frac {2}{\cos \left (2 \mathit {\_a} \mu \right )-1}\right )}{2}-\frac {a \ln \left (\cot ^{2}\left (\mu y \right )+1\right )}{2}+a \ln \left (\cot \left (\mu y \right )\right )\right ) \lambda }{b \mu }\right )\right )^{-\frac {c}{a \lambda }}}{\left (\sin \left (\frac {2 a \nu \ln \left (\frac {\cos \left (\mathit {\_a} \mu \right )}{\sin \left (\mathit {\_a} \mu \right )}\right )-a \nu \ln \left (-\frac {2}{\cos \left (2 \mathit {\_a} \mu \right )-1}\right )+a \nu \ln \left (\cot ^{2}\left (\mu y \right )+1\right )-2 a \nu \ln \left (\cot \left (\mu y \right )\right )-2 \left (-\mathit {\_a} \mu +\nu x \right ) b \mu }{2 b \mu }\right )+\sin \left (\frac {2 a \nu \ln \left (\frac {\cos \left (\mathit {\_a} \mu \right )}{\sin \left (\mathit {\_a} \mu \right )}\right )-a \nu \ln \left (-\frac {2}{\cos \left (2 \mathit {\_a} \mu \right )-1}\right )+a \nu \ln \left (\cot ^{2}\left (\mu y \right )+1\right )-2 a \nu \ln \left (\cot \left (\mu y \right )\right )-2 \left (\mathit {\_a} \mu +\nu x \right ) b \mu }{2 b \mu }\right )\right ) b}d\mathit {\_a} \right )+\mathit {\_F1} \left (\frac {2 b \mu x -a \ln \left (\cot ^{2}\left (\mu y \right )+1\right )+2 a \ln \left (\cot \left (\mu y \right )\right )}{2 b \mu }\right )\right ) \left (-\sin \left (\lambda x \right )\right )^{\frac {c}{a \lambda }}\]

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6.5.17.5 [1313] Problem 5

problem number 1313

Added April 11, 2019.

Problem Chapter 5.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c w + k \cot (\lambda x+\nu y) \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.17.6 [1314] Problem 6

problem number 1314

Added April 11, 2019.

Problem Chapter 5.6.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \cot ^n(\lambda x) w_x + b \cot ^m(\mu x) w_y = c \cot ^k(\nu x) w + p \cot ^s(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Cot[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Cot[nu*x]^k*w[x,y]+p*Cot[beta*y]^s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cot ^{-n}(\lambda K[2]) \cot ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cot ^{-n}(\lambda K[2]) \cot ^k(\nu K[2])}{a}dK[2]\right ) p \cot ^{-n}(\lambda K[3]) \cot ^s\left (\beta \left (y-\int _1^x\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*cot(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) =c*cot(nu*x)^k*w(x,y)+p*cot(beta*y)^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n} \left (\frac {\cos \left (\frac {\left (a y +b \left (\int \left (\frac {\cos \left (\mathit {\_f} \mu \right )}{\sin \left (\mathit {\_f} \mu \right )}\right )^{m} \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n}d \mathit {\_f} \right )-b \left (\int \left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right )^{-n} \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m}d x \right )\right ) \beta }{a}\right )}{\sin \left (\frac {\left (a y +b \left (\int \left (\frac {\cos \left (\mathit {\_f} \mu \right )}{\sin \left (\mathit {\_f} \mu \right )}\right )^{m} \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n}d \mathit {\_f} \right )-b \left (\int \left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right )^{-n} \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m}d x \right )\right ) \beta }{a}\right )}\right )^{s} {\mathrm e}^{-\frac {c \left (\int \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n} \left (\frac {\cos \left (\mathit {\_f} \nu \right )}{\sin \left (\mathit {\_f} \nu \right )}\right )^{k}d \mathit {\_f} \right )}{a}}}{a}d\mathit {\_f} +\mathit {\_F1} \left (\frac {a y -b \left (\int \left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right )^{-n} \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m}d x \right )}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\cot ^{-n}\left (\lambda x \right )\right ) \left (\cot ^{k}\left (\nu x \right )\right )}{a}d x}\]

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6.5.17.7 [1315] Problem 7

problem number 1315

Added April 11, 2019.

Problem Chapter 5.6.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \cot ^n(\lambda x) w_x + b \cot ^m(\mu x) w_y = c \cot ^k(\nu y) w + p \cot ^s(\beta x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Cot[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Cot[nu*y]^k*w[x,y]+p*Cot[beta*x]^s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cot ^{-n}(\lambda K[2]) \cot ^k\left (\nu \left (y-\int _1^x\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cot ^{-n}(\lambda K[2]) \cot ^k\left (\nu \left (y-\int _1^x\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) p \cot ^s(\beta K[3]) \cot ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cot ^{-n}(\lambda K[1]) \cot ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*cot(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) =c*cot(nu*y)^k*w(x,y)+p*cot(beta*x)^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (\frac {\cos \left (\mathit {\_f} \beta \right )}{\sin \left (\mathit {\_f} \beta \right )}\right )^{s} \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n} {\mathrm e}^{-\frac {c \left (\int \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n} \left (\frac {\cos \left (\frac {\left (a y +b \left (\int \left (\frac {\cos \left (\mathit {\_f} \mu \right )}{\sin \left (\mathit {\_f} \mu \right )}\right )^{m} \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n}d \mathit {\_f} \right )-b \left (\int \left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right )^{-n} \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m}d x \right )\right ) \nu }{a}\right )}{\sin \left (\frac {\left (a y +b \left (\int \left (\frac {\cos \left (\mathit {\_f} \mu \right )}{\sin \left (\mathit {\_f} \mu \right )}\right )^{m} \left (\frac {\cos \left (\mathit {\_f} \lambda \right )}{\sin \left (\mathit {\_f} \lambda \right )}\right )^{-n}d \mathit {\_f} \right )-b \left (\int \left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right )^{-n} \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m}d x \right )\right ) \nu }{a}\right )}\right )^{k}d \mathit {\_f} \right )}{a}}}{a}d\mathit {\_f} +\mathit {\_F1} \left (\frac {a y -b \left (\int \left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right )^{-n} \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m}d x \right )}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (\cot ^{-n}\left (\mathit {\_b} \lambda \right )\right ) \left (\cot ^{k}\left (\frac {\left (-b \left (\int \left (\frac {\cos \left (\lambda x \right )}{\sin \left (\lambda x \right )}\right )^{-n} \left (\frac {\cos \left (\mu x \right )}{\sin \left (\mu x \right )}\right )^{m}d x \right )+\left (y +\int \frac {b \left (\cot ^{-n}\left (\mathit {\_b} \lambda \right )\right ) \left (\cot ^{m}\left (\mathit {\_b} \mu \right )\right )}{a}d \mathit {\_b} \right ) a \right ) \nu }{a}\right )\right )}{a}d\mathit {\_b}}\]

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