Added April 11, 2019.
Problem Chapter 5.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \sin ^k(\lambda x)+c_2 \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+c1*Sin[lambda*x]^k+c2*Cos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} c_1\left (y-\frac {b x}{a}\right )-\frac {i \text {c1} \left (-1+e^{2 i \lambda x}\right ) \sin ^k(\lambda x) \, _2F_1\left (1,\frac {1}{2} \left (k+\frac {i}{a \lambda }+2\right );\frac {1}{2} \left (-k+\frac {i}{a \lambda }+2\right );e^{2 i \lambda x}\right )}{a k \lambda -i}+\frac {\text {c2} \left (1+e^{2 i \beta y}\right ) \cos ^n(\beta y) \, _2F_1\left (1,\frac {1}{2} \left (n+\frac {i}{b \beta }+2\right );\frac {1}{2} \left (-n+\frac {i}{b \beta }+2\right );-e^{2 i \beta y}\right )}{-1-i b \beta n}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*sin(lambda*x)^k+c2*cos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \sin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+{\it c2}\, \left ( \cos \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n} \right ) {{\rm e}^{-{\frac {{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \sin ^k(\lambda x) \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+Sin[lambda*x]^k*Cos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cos ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \sin ^k(\lambda K[1])}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+sin(lambda*x)^k*cos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \sin \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \cos \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin (\mu y) w_y = c \sin (\lambda x) w + k \cos (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Sin[mu*y]*D[w[x, y], y] == c*Sin[lambda*x]*w[x,y]+k*Cos[nu*x]+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {c \cos (\lambda x)}{a \lambda }} \left (\int _1^x\frac {e^{\frac {c \cos (\lambda K[1])}{a \lambda }} (s+k \cos (\nu K[1]))}{a}dK[1]+c_1\left (\frac {\log \left (\tan \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*sin(mu*y)*diff(w(x,y),y) = c*sin(lambda*x)*w(x,y)+k*cos(nu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {k\cos \left ( \nu \,x \right ) +s}{a}{{\rm e}^{{\frac {c\cos \left ( x\lambda \right ) }{a\lambda }}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {a}{\mu \,b}\ln \left ( \RootOf \left ( \mu \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {b\mu \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \right ) {{\rm e}^{-{\frac {c\cos \left ( x\lambda \right ) }{a\lambda }}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \sin (\mu y) w_y = c \sin (\lambda x) w + k \tan (\nu x) + s \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*Sin[mu*y]*D[w[x, y], y] == c*Sin[lambda*x]*w[x,y]+k*Tan[nu*x]+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{-\frac {c \cos (\lambda x)}{a \lambda }} \left (\int _1^x\frac {e^{\frac {c \cos (\lambda K[1])}{a \lambda }} (s+k \tan (\nu K[1]))}{a}dK[1]+c_1\left (\frac {\log \left (\tan \left (\frac {\mu y}{2}\right )\right )}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*sin(mu*y)*diff(w(x,y),y) = c*sin(lambda*x)*w(x,y)+k*tan(nu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {k\sin \left ( \nu \,x \right ) +s\cos \left ( \nu \,x \right ) }{\cos \left ( \nu \,x \right ) a}{{\rm e}^{{\frac {c\cos \left ( x\lambda \right ) }{a\lambda }}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {a}{\mu \,b}\ln \left ( \RootOf \left ( \mu \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {b\mu \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\mu \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \right ) {{\rm e}^{-{\frac {c\cos \left ( x\lambda \right ) }{a\lambda }}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \tan (\mu y) w_y = c \tan (\lambda x) w + k \cot (\nu x) + s \]
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*Cot[nu*x]+s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \cos ^{-\frac {c}{a \lambda }}(\lambda x) \left (\int _1^x\frac {\cos ^{\frac {c}{a \lambda }}(\lambda K[1]) (s+k \cot (\nu K[1]))}{a}dK[1]+c_1\left (\frac {\log (\sin (\mu y))}{\mu }-\frac {b x}{a}\right )\right )\right \}\right \}\]
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*cot(nu*x)+s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {\sin \left ( \nu \,x \right ) s+k\cos \left ( \nu \,x \right ) }{\sin \left ( \nu \,x \right ) a} \left ( \cos \left ( x\lambda \right ) \right ) ^{{\frac {c}{a\lambda }}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{\mu \,b} \left ( -b\mu \,x+\ln \left ( {\tan \left ( \mu \,y \right ) {\frac {1}{\sqrt {1+ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}}}}} \right ) a \right ) } \right ) \right ) \left ( \cos \left ( x\lambda \right ) \right ) ^{-{\frac {c}{a\lambda }}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \sin ^n(\lambda x) w_x + b \cos ^m(\mu x) w_y = c \cos ^k(\nu x) w + p \sin ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Sin[lambda*x]^n*D[w[x, y], x] + b*Cos[mu*x]^m*D[w[x, y], y] == c*Cos[nu*x]^k*w[x,y]+p*Sin[beta*y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cos ^k(\nu K[2]) \sin ^{-n}(\lambda K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cos ^k(\nu K[2]) \sin ^{-n}(\lambda K[2])}{a}dK[2]\right ) p \sin ^{-n}(\lambda K[3]) \sin ^s\left (\beta \left (y-\int _1^x\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cos ^m(\mu K[1]) \sin ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*sin(lambda*x)^n*diff(w(x,y),x)+ b*cos(mu*x)^m*diff(w(x,y),y) = c*cos(nu*x)^k*w(x,y)+p*sin(beta*y)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \sin \left ( {\frac {\beta \, \left ( b\int \! \left ( \cos \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}+ay-b\int \! \left ( \cos \left ( \mu \,x \right ) \right ) ^{m} \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x \right ) }{a}} \right ) \right ) ^{s}{{\rm e}^{-{\frac {c\int \! \left ( \cos \left ( \nu \,{\it \_f} \right ) \right ) ^{k} \left ( \sin \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}{a}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {ay-b\int \! \left ( \cos \left ( \mu \,x \right ) \right ) ^{m} \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x}{a}} \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \cos \left ( \nu \,x \right ) \right ) ^{k}c \left ( \sin \left ( x\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}x}}\]
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Added April 11, 2019.
Problem Chapter 5.6.5.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a \tan ^n(\lambda x) w_x + b \cot ^m(\mu x) w_y = c \tan ^k(\nu x) w + p \cot ^s(\beta x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Tan[lambda*x]^n*D[w[x, y], x] + b*Cot[mu*x]^m*D[w[x, y], y] == c*Tan[nu*x]^k*w[x,y]+p*Cot[beta*x]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \tan ^{-n}(\lambda K[2]) \tan ^k(\nu K[2])}{a}dK[2]\right ) p \cot ^s(\beta K[3]) \tan ^{-n}(\lambda K[3])}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cot ^m(\mu K[1]) \tan ^{-n}(\lambda K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*tan(lambda*x)^n*diff(w(x,y),x)+ b*cot(mu*x)^m*diff(w(x,y),y) = c*tan(nu*x)^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'));
\[w \left ( x,y \right ) = \left ( \int \!{\frac {p}{a} \left ( {\frac {\cos \left ( \beta \,x \right ) }{\sin \left ( \beta \,x \right ) }} \right ) ^{s} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {\frac {\sin \left ( \nu \,x \right ) }{\cos \left ( \nu \,x \right ) }} \right ) ^{k} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x}}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{a} \left ( -b\int \! \left ( {\frac {\cos \left ( \mu \,x \right ) }{\sin \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sin \left ( x\lambda \right ) }{\cos \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x+ay \right ) } \right ) \right ) {{\rm e}^{\int \!{\frac {c \left ( \tan \left ( x\lambda \right ) \right ) ^{-n}}{a} \left ( {\frac {\sin \left ( \nu \,x \right ) }{\cos \left ( \nu \,x \right ) }} \right ) ^{k}}\,{\rm d}x}}\]
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