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'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}\cot \left ( {\frac { \left ( {\it \_a}\,\lambda +\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) {{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]

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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'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c1}\, \left ( \cot \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+{\it c2}\, \left ( \cot \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]

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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'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \cot \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \cot \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]

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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 ( \int ^{y}\!{\frac {k}{b} \left ( -\sin \left ( {\frac {\lambda }{2\,\mu \,b} \left ( 2\,b\mu \,x+2\,a\ln \left ( \cot \left ( \mu \,y \right ) \right ) -a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) +a\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) -2\,a\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) \right ) } \right ) \right ) ^{-{\frac {c}{a\lambda }}} \left ( \sin \left ( {\frac {1}{2\,\mu \,b} \left ( \ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu -2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu -a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \nu +2\,a\ln \left ( \cot \left ( \mu \,y \right ) \right ) \nu +2\,b\mu \, \left ( \mu \,{\it \_a}+\nu \,x \right ) \right ) } \right ) -\sin \left ( {\frac {1}{2\,\mu \,b} \left ( \left ( 2\,b\mu \,x+2\,a\ln \left ( \cot \left ( \mu \,y \right ) \right ) -a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \right ) \nu -2\,{\mu }^{2}{\it \_a}\,b+\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu -2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu \right ) } \right ) \right ) \left ( \sin \left ( {\frac {1}{2\,\mu \,b} \left ( \ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu -2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu -a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \nu +2\,a\ln \left ( \cot \left ( \mu \,y \right ) \right ) \nu +2\,b\mu \, \left ( \mu \,{\it \_a}+\nu \,x \right ) \right ) } \right ) +\sin \left ( {\frac {1}{2\,\mu \,b} \left ( \left ( 2\,b\mu \,x+2\,a\ln \left ( \cot \left ( \mu \,y \right ) \right ) -a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \right ) \nu -2\,{\mu }^{2}{\it \_a}\,b+\ln \left ( -2\, \left ( -1+\cos \left ( 2\,\mu \,{\it \_a} \right ) \right ) ^{-1} \right ) a\nu -2\,\ln \left ( {\frac {\cos \left ( \mu \,{\it \_a} \right ) }{\sin \left ( \mu \,{\it \_a} \right ) }} \right ) a\nu \right ) } \right ) \right ) ^{-1}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {2\,b\mu \,x+2\,a\ln \left ( \cot \left ( \mu \,y \right ) \right ) -a\ln \left ( \left ( \cot \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) }{2\,\mu \,b}} \right ) \right ) \left ( -\sin \left ( x\lambda \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'));
 

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

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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}{a} \left ( {\cos \left ( {\frac {\beta }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,{\it \_f} \right ) }{\sin \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\cos \left ( x\lambda \right ) }{\sin \left ( x\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m}\,{\rm d}x+ay \right ) } \right ) \left ( \sin \left ( {\frac {\beta }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,{\it \_f} \right ) }{\sin \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\cos \left ( x\lambda \right ) }{\sin \left ( x\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m}\,{\rm d}x+ay \right ) } \right ) \right ) ^{-1}} \right ) ^{s} \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \nu \,{\it \_f} \right ) }{\sin \left ( \nu \,{\it \_f} \right ) }} \right ) ^{k}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{a} \left ( -b\int \! \left ( {\frac {\cos \left ( x\lambda \right ) }{\sin \left ( x\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m}\,{\rm d}x+ay \right ) } \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \cot \left ( \nu \,x \right ) \right ) ^{k}c \left ( \cot \left ( x\lambda \right ) \right ) ^{-n}}{a}}\,{\rm 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}{a} \left ( {\frac {\cos \left ( \beta \,{\it \_f} \right ) }{\sin \left ( \beta \,{\it \_f} \right ) }} \right ) ^{s} \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {\cos \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,{\it \_f} \right ) }{\sin \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\cos \left ( x\lambda \right ) }{\sin \left ( x\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m}\,{\rm d}x+ay \right ) } \right ) \left ( \sin \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,{\it \_f} \right ) }{\sin \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\cos \left ( x\lambda \right ) }{\sin \left ( x\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m}\,{\rm d}x+ay \right ) } \right ) \right ) ^{-1}} \right ) ^{k} \left ( {\frac {\cos \left ( {\it \_f}\,\lambda \right ) }{\sin \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{a} \left ( -b\int \! \left ( {\frac {\cos \left ( x\lambda \right ) }{\sin \left ( x\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m}\,{\rm d}x+ay \right ) } \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \cot \left ( {\frac {\nu }{a} \left ( -b\int \! \left ( {\frac {\cos \left ( x\lambda \right ) }{\sin \left ( x\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m}\,{\rm d}x+a \left ( \int \!{\frac {b \left ( \cot \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}{\it \_b}+y \right ) \right ) } \right ) \right ) ^{k}}{d{\it \_b}}}}\]

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