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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {k \,{\mathrm e}^{-\frac {\textit {\_a} c}{a}} \tan \left (\frac {-\left (-\textit {\_a} +x \right ) b \mu +\left (\textit {\_a} \lambda +\mu y \right ) a}{a}\right )}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \left (\tan ^{k}\left (\textit {\_a} \lambda \right )\right )+\mathit {c2} \left (\tan ^{n}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right )\right ) {\mathrm e}^{-\frac {\textit {\_a}}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\tan ^{k}\left (\textit {\_a} \lambda \right )\right ) \left (\tan ^{n}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {\textit {\_a} c}{a}}}{a}d \textit {\_a} +\textit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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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'));
\[w \left (x , y\right ) = \left (\int \frac {k \left (\frac {2}{\cos \left (2 \lambda x \right )+1}\right )^{-\frac {c}{2 a \lambda }} \sin \left (\nu x \right )}{a \cos \left (\nu x \right )}d x +\textit {\_F1} \left (\frac {-b \mu x +a \ln \left (\frac {\tan \left (\mu y \right )}{\sqrt {\tan ^{2}\left (\mu y \right )+1}}\right )}{b \mu }\right )\right ) \left (\tan ^{2}\left (\lambda x \right )+1\right )^{\frac {c}{2 a \lambda }}\]
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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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {k \,\textit {\_a}^{\frac {-a -c}{a}} \tan \left (\nu y \,\textit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}+\textit {\_a} \lambda \right )}{a}d \textit {\_a} +\textit {\_F1} \left (y \,x^{-\frac {b}{a}}\right )\right ) x^{\frac {c}{a}}\]
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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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n} \left (\frac {\sin \left (\frac {\left (a y +b \left (\int \left (\frac {\sin \left (\textit {\_f} \mu \right )}{\cos \left (\textit {\_f} \mu \right )}\right )^{m} \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n}d \textit {\_f} \right )-b \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\mu x \right )}{\cos \left (\mu x \right )}\right )^{m}d x \right )\right ) \beta }{a}\right )}{\cos \left (\frac {\left (a y +b \left (\int \left (\frac {\sin \left (\textit {\_f} \mu \right )}{\cos \left (\textit {\_f} \mu \right )}\right )^{m} \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n}d \textit {\_f} \right )-b \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\mu x \right )}{\cos \left (\mu x \right )}\right )^{m}d x \right )\right ) \beta }{a}\right )}\right )^{s} {\mathrm e}^{-\frac {c \left (\int \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n} \left (\frac {\sin \left (\textit {\_f} \nu \right )}{\cos \left (\textit {\_f} \nu \right )}\right )^{k}d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\mu x \right )}{\cos \left (\mu x \right )}\right )^{m}d x \right )}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\tan ^{-n}\left (\lambda x \right )\right ) \left (\tan ^{k}\left (\nu x \right )\right )}{a}d x}\]
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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 \left (\frac {\sin \left (\textit {\_f} \beta \right )}{\cos \left (\textit {\_f} \beta \right )}\right )^{s} \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n} {\mathrm e}^{-\frac {c \left (\int \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n} \left (\frac {\sin \left (\frac {\left (a y +b \left (\int \left (\frac {\sin \left (\textit {\_f} \mu \right )}{\cos \left (\textit {\_f} \mu \right )}\right )^{m} \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n}d \textit {\_f} \right )-b \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\mu x \right )}{\cos \left (\mu x \right )}\right )^{m}d x \right )\right ) \nu }{a}\right )}{\cos \left (\frac {\left (a y +b \left (\int \left (\frac {\sin \left (\textit {\_f} \mu \right )}{\cos \left (\textit {\_f} \mu \right )}\right )^{m} \left (\frac {\sin \left (\textit {\_f} \lambda \right )}{\cos \left (\textit {\_f} \lambda \right )}\right )^{-n}d \textit {\_f} \right )-b \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\mu x \right )}{\cos \left (\mu x \right )}\right )^{m}d x \right )\right ) \nu }{a}\right )}\right )^{k}d \textit {\_f} \right )}{a}}}{a}d \textit {\_f} +\textit {\_F1} \left (\frac {a y -b \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\mu x \right )}{\cos \left (\mu x \right )}\right )^{m}d x \right )}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (\tan ^{-n}\left (\textit {\_b} \lambda \right )\right ) \left (\tan ^{k}\left (\frac {\left (-b \left (\int \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-n} \left (\frac {\sin \left (\mu x \right )}{\cos \left (\mu x \right )}\right )^{m}d x \right )+\left (y +\int \frac {b \left (\tan ^{-n}\left (\textit {\_b} \lambda \right )\right ) \left (\tan ^{m}\left (\textit {\_b} \mu \right )\right )}{a}d \textit {\_b} \right ) a \right ) \nu }{a}\right )\right )}{a}d \textit {\_b}}\]
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