Added April 11, 2019.
Problem Chapter 5.6.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + k \tan (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+k*Tan[lambda*x+mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )-\frac {i k \left (-2 \, _2F_1\left (1,-\frac {i c}{2 a \lambda +2 b \mu };\frac {-i c+2 a \lambda +2 b \mu }{2 (a \lambda +b \mu )};-e^{-2 i (\lambda x+\mu y)}\right )+2 e^{\frac {2 i \mu (a y-b x)}{a}} \, _2F_1\left (1,\frac {i c}{2 a \lambda +2 b \mu };\frac {i c+2 a \lambda +2 b \mu }{2 a \lambda +2 b \mu };-e^{2 i (\lambda x+\mu y)}\right )-e^{\frac {2 i \mu (a y-b x)}{a}}+1\right )}{c \left (1+e^{\frac {2 i \mu (a y-b x)}{a}}\right )}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) =c*w(x,y)+k*tan(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}\tan \left ( {\frac { \left ( {\it \_a}\,\lambda +\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) {{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = w + c_1 \tan ^k(\lambda x) + c_2 \tan ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*Tan[lambda*x]^k + c2*Tan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \tan ^k(\lambda K[1])+\text {c2} \tan ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+ c1*tan(lambda*x)^k + c2*tan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {{\it \_a}}{a}}}} \left ( {\it c2}\, \left ( \tan \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}+{\it c1}\, \left ( \tan \left ( {\it \_a}\,\lambda \right ) \right ) ^{k} \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + \tan ^k(\lambda x) \tan ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ Tan[lambda*x]^k * Tan[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \tan ^k(\lambda K[1]) \tan ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ tan(lambda*x)^k *tan(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \tan \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \tan \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b \tan (\mu y) w_y = c \tan (\lambda x) w + k \tan (\nu x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Tan[mu*y]*D[w[x, y], y] == c*Tan[lambda*x]*w[x,y]+k*Tan[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \cos ^{-\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x\frac {k \cos ^{\frac {c}{a \lambda }}(\lambda K[1]) \tan (\nu K[1])}{a}dK[1]+c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*tan(mu*y)*diff(w(x,y),y) = c*tan(lambda*x)*w(x,y)+ k*tan(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {k\sin \left ( \nu \,x \right ) }{a\cos \left ( \nu \,x \right ) } \left ( 2\, \left ( \cos \left ( 2\,x\lambda \right ) +1 \right ) ^{-1} \right ) ^{-{\frac {c}{2\,a\lambda }}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{\mu \,b} \left ( -b\mu \,x+\ln \left ( {\tan \left ( \mu \,y \right ) {\frac {1}{\sqrt {1+ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}}}}} \right ) a \right ) } \right ) \right ) \left ( 1+ \left ( \tan \left ( x\lambda \right ) \right ) ^{2} \right ) ^{{\frac {c}{2\,a\lambda }}}\]
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Added April 11, 2019.
Problem Chapter 5.6.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x w_x + b y w_y = c w + k \tan (\lambda x+\nu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*w[x,y]+k*Tan[lambda*x+nu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to x^{\frac {c}{a}} \left (\int _1^x\frac {k K[1]^{-\frac {a+c}{a}} \tan \left (\nu y K[1]^{\frac {b}{a}} x^{-\frac {b}{a}}+\lambda K[1]\right )}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) =c*w(x,y)+k*tan(lambda*x+nu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}\tan \left ( {\it \_a}\,\lambda +\nu \,y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) {{\it \_a}}^{{\frac {-a-c}{a}}}}{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {x}^{{\frac {c}{a}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tan ^n(\lambda x) w_x + b \tan ^m(\mu x) w_y = c \tan ^k(\nu x) w + p \tan ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Tan[lambda*x]^n*D[w[x, y], x] + b*Tan[mu*x]^m*D[w[x, y], y] == c*Tan[nu*x]^k*w[x,y]+p*Tan[beta*y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) p \tan ^{-n}(\lambda K[3]) \tan ^s\left (\beta \left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*tan(lambda*x)^n*diff(w(x,y),x)+ b*tan(mu*x)^m*diff(w(x,y),y) =c*tan(nu*x)^k*w(x,y)+p*tan(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 ( {\sin \left ( {\frac {\beta }{a} \left ( b\int \! \left ( {\frac {\sin \left ( \mu \,{\it \_f} \right ) }{\cos \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\sin \left ( \mu \,x \right ) }{\cos \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \left ( \cos \left ( {\frac {\beta }{a} \left ( b\int \! \left ( {\frac {\sin \left ( \mu \,{\it \_f} \right ) }{\cos \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\sin \left ( \mu \,x \right ) }{\cos \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) ^{-1}} \right ) ^{s} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {\frac {\sin \left ( \nu \,{\it \_f} \right ) }{\cos \left ( \nu \,{\it \_f} \right ) }} \right ) ^{k} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \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 {\sin \left ( \mu \,x \right ) }{\cos \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \tan \left ( \nu \,x \right ) \right ) ^{k}c \left ( \tan \left ( x\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}x}}\]
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Added April 11, 2019.
Problem Chapter 5.6.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tan ^n(\lambda x) w_x + b \tan ^m(\mu x) w_y = c \tan ^k(\nu y) w + p \tan ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Tan[lambda*x]^n*D[w[x, y], x] + b*Tan[mu*x]^m*D[w[x, y], y] == c*Tan[nu*y]^k*w[x,y]+p*Tan[beta*x]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k\left (\nu \left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k\left (\nu \left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[2]\right ) p \tan ^s(\beta K[3]) \tan ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \tan ^{-n}(\lambda K[1]) \tan ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*tan(lambda*x)^n*diff(w(x,y),x)+ b*tan(mu*x)^m*diff(w(x,y),y) =c*tan(nu*y)^k*w(x,y)+p*tan(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 {\sin \left ( \beta \,{\it \_f} \right ) }{\cos \left ( \beta \,{\it \_f} \right ) }} \right ) ^{s} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {\sin \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\sin \left ( \mu \,{\it \_f} \right ) }{\cos \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\sin \left ( \mu \,x \right ) }{\cos \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \left ( \cos \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\sin \left ( \mu \,{\it \_f} \right ) }{\cos \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}-b\int \! \left ( {\frac {\sin \left ( \mu \,x \right ) }{\cos \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) ^{-1}} \right ) ^{k} \left ( {\frac {\sin \left ( {\it \_f}\,\lambda \right ) }{\cos \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 {\sin \left ( \mu \,x \right ) }{\cos \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( \tan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}c}{a} \left ( \tan \left ( {\frac {\nu }{a} \left ( -b\int \! \left ( {\frac {\sin \left ( \mu \,x \right ) }{\cos \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+a \left ( \int \!{\frac {b \left ( \tan \left ( \mu \,{\it \_b} \right ) \right ) ^{m} \left ( \tan \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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