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'));
\[w \left (x , y\right ) = \frac {\left (\left (-c \cos \left (\lambda x +\mu y \right )+\left (a \lambda +\mu b \right ) \sin \left (\lambda x +\mu y \right )\right ) k \,{\mathrm e}^{-\frac {c x}{a}}+\left (a^{2} \lambda ^{2}+2 a b \lambda \mu +b^{2} \mu ^{2}+c^{2}\right ) \mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}}{a^{2} \lambda ^{2}+2 a b \lambda \mu +b^{2} \mu ^{2}+c^{2}}\]
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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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \left (\cos ^{k}\left (\mathit {\_a} \lambda \right )\right )+\mathit {c2} \left (\cos ^{n}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )\right )\right ) {\mathrm e}^{-\frac {\mathit {\_a}}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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Added April 11, 2019.
Problem Chapter 5.6.2.3, 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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\cos ^{k}\left (\mathit {\_a} \lambda \right )\right ) \left (\cos ^{n}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {\mathit {\_a} c}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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Added April 11, 2019.
Problem Chapter 5.6.2.4, 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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {k \mathit {\_a}^{\frac {-a -c}{a}} \cos \left (\mu y \mathit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}+\mathit {\_a} \lambda \right )}{a}d\mathit {\_a} +\mathit {\_F1} \left (y x^{-\frac {b}{a}}\right )\right ) x^{\frac {c}{a}}\]
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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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {b \cos \left (\mathit {\_a} \nu \right ) {\mathrm e}^{-\frac {a \sin \left (\mathit {\_a} \lambda +\frac {\mathit {\_a} \mu y}{x}\right )}{\lambda +\frac {\mu y}{x}}}}{\mathit {\_a}}d\mathit {\_a} +\mathit {\_F1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{\frac {a \sin \left (\lambda x +\mu y \right )}{\lambda +\frac {\mu y}{x}}}\]
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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'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {p \left (\cos ^{-n}\left (\mathit {\_f} \lambda \right )\right ) \left (\cos ^{s}\left (\frac {\left (a y +b \left (\int \left (\cos ^{m}\left (\mathit {\_f} \mu \right )\right ) \left (\cos ^{-n}\left (\mathit {\_f} \lambda \right )\right )d \mathit {\_f} \right )-b \left (\int \left (\cos ^{-n}\left (\lambda x \right )\right ) \left (\cos ^{m}\left (\mu x \right )\right )d x \right )\right ) \beta }{a}\right )\right ) {\mathrm e}^{-\frac {c \left (\int \left (\cos ^{-n}\left (\mathit {\_f} \lambda \right )\right ) \left (\cos ^{k}\left (\mathit {\_f} \nu \right )\right )d \mathit {\_f} \right )}{a}}}{a}d\mathit {\_f} +\mathit {\_F1} \left (\frac {a y -b \left (\int \left (\cos ^{-n}\left (\lambda x \right )\right ) \left (\cos ^{m}\left (\mu x \right )\right )d x \right )}{a}\right )\right ) {\mathrm e}^{\int \frac {c \left (\cos ^{-n}\left (\lambda x \right )\right ) \left (\cos ^{k}\left (\nu x \right )\right )}{a}d x}\]
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