Added April 11, 2019.
Problem Chapter 5.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + k \cos (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ k*Cos[lambda*x+mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {k ((a \lambda +b \mu ) \sin (\lambda x+\mu y)-c \cos (\lambda x+\mu y))}{(a \lambda +b \mu )^2+c^2}+e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ k*cos(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\frac {1}{{a}^{2}{\lambda }^{2}+2\,ab\lambda \,\mu +{b}^{2}{\mu }^{2}+{c}^{2}}{{\rm e}^{{\frac {cx}{a}}}} \left ( \left ( {a}^{2}{\lambda }^{2}+2\,ab\lambda \,\mu +{b}^{2}{\mu }^{2}+{c}^{2} \right ) {\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) +k \left ( \left ( a\lambda +\mu \,b \right ) \sin \left ( x\lambda +\mu \,y \right ) -c\cos \left ( x\lambda +\mu \,y \right ) \right ) {{\rm e}^{-{\frac {cx}{a}}}} \right ) }\]
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Added April 11, 2019.
Problem Chapter 5.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = w + c_1 \cos ^k(\lambda x) + c_2 \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*Cos[lambda*x]^k + c2*Cos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \frac {e^{\frac {x}{a}} (a k \lambda -i) (b \beta n-i) c_1\left (y-\frac {b x}{a}\right )+\text {c1} \left (1+e^{2 i \lambda x}\right ) (1+i b \beta n) \cos ^k(\lambda x) \, _2F_1\left (1,\frac {1}{2} \left (k+\frac {i}{a \lambda }+2\right );\frac {1}{2} \left (-k+\frac {i}{a \lambda }+2\right );-e^{2 i \lambda x}\right )+\text {c2} \left (1+e^{2 i \beta y}\right ) (1+i a k \lambda ) \cos ^n(\beta y) \, _2F_1\left (1,\frac {1}{2} \left (n+\frac {i}{b \beta }+2\right );\frac {1}{2} \left (-n+\frac {i}{b \beta }+2\right );-e^{2 i \beta y}\right )}{(a k \lambda -i) (b \beta n-i)}\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+ c1*cos(lambda*x)^k + c2*cos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \cos \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+{\it c2}\, \left ( \cos \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n} \right ) {{\rm e}^{-{\frac {{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a w_x + b w_y = c w + \cos ^k(\lambda x) \cos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ Cos[lambda*x]^k * Cos[beta*y]^n; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \cos ^k(\lambda K[1]) \cos ^n\left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ cos(lambda*x)^k *cos(beta*y)^n; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \cos \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \cos \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a x w_x + b y w_y = c w + k \cos (\lambda x+\mu y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*w[x,y]+ k*Cos[lambda*x+mu*y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to x^{\frac {c}{a}} \left (\int _1^x\frac {k \cos \left (\mu y K[1]^{\frac {b}{a}} x^{-\frac {b}{a}}+\lambda K[1]\right ) K[1]^{-\frac {a+c}{a}}}{a}dK[1]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*w(x,y)+ k*cos(lambda*x+mu*y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}{{\it \_a}}^{{\frac {-a-c}{a}}}\cos \left ( {\it \_a}\,\lambda +\mu \,y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) }{d{\it \_a}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {x}^{{\frac {c}{a}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \cos (\lambda x+\mu y) w + b \cos (\nu x) \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Cos[lambda*x+mu*y]*w[x,y]+b*Cos[nu*x]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {a x \sin (\lambda x+\mu y)}{\lambda x+\mu y}} \left (\int _1^x\frac {b \exp \left (-\frac {a x \sin \left (\left (\lambda +\frac {\mu y}{x}\right ) K[1]\right )}{\lambda x+\mu y}\right ) \cos (\nu K[1])}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) =a*x*cos(lambda*x+mu*y)*w(x,y)+b*cos(nu*x); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {\cos \left ( \nu \,{\it \_a} \right ) b}{{\it \_a}}{{\rm e}^{-{a\sin \left ( {\frac {\mu \,y{\it \_a}}{x}}+{\it \_a}\,\lambda \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) {{\rm e}^{{a\sin \left ( x\lambda +\mu \,y \right ) \left ( {\frac {\mu \,y}{x}}+\lambda \right ) ^{-1}}}}\]
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Added April 11, 2019.
Problem Chapter 5.6.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ a \cos ^n(\lambda x) w_x + b \cos ^m(\mu x) w_y = c \cos ^k(\nu x) w + p \cos ^s(\beta y) \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*Cos[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*Cos[beta*y]^s; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \cos ^{-n}(\lambda K[2]) \cos ^k(\nu K[2])}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \cos ^{-n}(\lambda K[2]) \cos ^k(\nu K[2])}{a}dK[2]\right ) p \cos ^{-n}(\lambda K[3]) \cos ^s\left (\beta \left (y-\int _1^x\frac {b \cos ^{-n}(\lambda K[1]) \cos ^m(\mu K[1])}{a}dK[1]+\int _1^{K[3]}\frac {b \cos ^{-n}(\lambda K[1]) \cos ^m(\mu K[1])}{a}dK[1]\right )\right )}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \cos ^{-n}(\lambda K[1]) \cos ^m(\mu K[1])}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := a*cos(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*cos(beta*y)^s; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {p \left ( \cos \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}}{a} \left ( \cos \left ( {\frac {\beta \, \left ( b\int \! \left ( \cos \left ( \mu \,{\it \_f} \right ) \right ) ^{m} \left ( \cos \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}+ay-b\int \! \left ( \cos \left ( \mu \,x \right ) \right ) ^{m} \left ( \cos \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x \right ) }{a}} \right ) \right ) ^{s}{{\rm e}^{-{\frac {c\int \! \left ( \cos \left ( \nu \,{\it \_f} \right ) \right ) ^{k} \left ( \cos \left ( {\it \_f}\,\lambda \right ) \right ) ^{-n}\,{\rm d}{\it \_f}}{a}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {ay-b\int \! \left ( \cos \left ( \mu \,x \right ) \right ) ^{m} \left ( \cos \left ( x\lambda \right ) \right ) ^{-n}\,{\rm d}x}{a}} \right ) \right ) {{\rm e}^{\int \!{\frac { \left ( \cos \left ( \nu \,x \right ) \right ) ^{k}c \left ( \cos \left ( x\lambda \right ) \right ) ^{-n}}{a}}\,{\rm d}x}}\]
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