Added January 14, 2019.
Problem 2.6.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \tan ^k(\lambda x)+b\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a + Tan[lambda*x] + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-a x-b x+\frac {\log (\cos (\lambda x))}{\lambda }+y\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(a+tan(lambda*x)+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (\left (-2 a -2 b \right ) x +2 y \right ) \lambda -\ln \left (\tan ^{2}\left (\lambda x \right )+1\right )}{2 \lambda }\right )\]
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Added January 14, 2019.
Problem 2.6.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + \left ( a \tan ^k(\lambda y)+b\right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a + Tan[lambda*y] + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-x+\frac {-i (a+b-i) \log (-\tan (\lambda y)+i)+i (a+b+i) \log (\tan (\lambda y)+i)+2 \log (a+b+\tan (\lambda y))}{2 \lambda (a+b-i) (a+b+i)}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+(a+tan(lambda*y)+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (2 a^{2} x +4 a b x +2 b^{2} x +2 x \right ) \lambda +\left (-2 a -2 b \right ) \arctan \left (\tan \left (\lambda y \right )\right )+\ln \left (\tan ^{2}\left (\lambda y \right )+1\right )-2 \ln \left (a +b +\tan \left (\lambda y \right )\right )}{2 \left (a^{2}+2 a b +b^{2}+1\right ) \lambda }\right )\]
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Added January 14, 2019.
Problem 2.6.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x + a \tan ^k(\lambda x) \tan ^n(\mu y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*Tan[lambda*x]^k*Tan[mu*y]^n*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {\tan ^{1-n}(\mu y) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};-\tan ^2(\mu y)\right )}{\mu -\mu n}-\frac {a \tan ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\tan ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+a *tan(lambda*x)^k * tan(mu*y)^n*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (-\left (\int \left (\tan ^{k}\left (\lambda x \right )\right )d x \right )+\int \frac {\tan ^{-n}\left (\mu y \right )}{a}d y \right )\] Has unresolved integrals
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Added January 14, 2019.
Problem 2.6.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( y^2+ a \lambda + a(\lambda -a) \tan ^2(\lambda x)\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*lambda + a*(lambda - a)*Tan[lambda*x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+( y^2+ a *lambda + a*(lambda -a) *tan(lambda*x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {2 \left (\cos \left (2 \lambda x \right )+1\right ) \lambda \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )+\left (-4 y \left (\cos ^{3}\left (\lambda x \right )\right )-a \sin \left (\lambda x \right )-a \sin \left (3 \lambda x \right )\right ) \LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )}{-2 \left (\cos \left (2 \lambda x \right )+1\right ) \lambda \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )+\left (4 y \left (\cos ^{3}\left (\lambda x \right )\right )+a \sin \left (\lambda x \right )+a \sin \left (3 \lambda x \right )\right ) \LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )}\right )\]
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Added January 14, 2019.
Problem 2.6.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( y^2+ \lambda ^2 +3 a \lambda +a(\lambda -a) \tan ^2(\lambda x)\right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + lambda^2 + 3*a*lambda + a*(lambda - a)*Tan[lambda*x]^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)+( y^2+ lambda^2 +3*a*lambda +a*(lambda-a)*tan(lambda*x)^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {4 \lambda \LegendreP \left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) \left (\cos ^{2}\left (\lambda x \right )\right )-2 \left (\frac {\lambda \left (\sin ^{3}\left (\lambda x \right )\right )}{2}+\left (a +\frac {3 \lambda }{2}\right ) \left (\cos ^{2}\left (\lambda x \right )\right ) \sin \left (\lambda x \right )-\frac {\lambda \sin \left (\lambda x \right )}{2}+\left (-y \left (\sin ^{2}\left (\lambda x \right )\right )+y \right ) \cos \left (\lambda x \right )\right ) \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )}{-4 \lambda \LegendreQ \left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) \left (\cos ^{2}\left (\lambda x \right )\right )+2 \left (\frac {\lambda \left (\sin ^{3}\left (\lambda x \right )\right )}{2}+\left (a +\frac {3 \lambda }{2}\right ) \left (\cos ^{2}\left (\lambda x \right )\right ) \sin \left (\lambda x \right )-\frac {\lambda \sin \left (\lambda x \right )}{2}+\left (-y \left (\sin ^{2}\left (\lambda x \right )\right )+y \right ) \cos \left (\lambda x \right )\right ) \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )}\right )\]
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Added January 14, 2019.
Problem 2.6.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( y^2+ a x \tan ^k(b x) y + a \tan ^k(b x) \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*x*Tan[b*x]^k*y + a*Tan[b*x]^k)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\exp \left (-\int _1^x-a K[5] \tan ^k(b K[5])dK[5]\right )}{x^2 y+x}-\int _1^x\frac {\exp \left (-\int _1^{K[6]}-a K[5] \tan ^k(b K[5])dK[5]\right )}{K[6]^2}dK[6]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+( y^2+ a*x *tan(b*x)^k * y + a*tan(b*x)^k)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {x y \left (\int {\mathrm e}^{\int \frac {a x^{2} \left (\tan ^{k}\left (b x \right )\right )-2}{x}d x}d x \right )+x \,{\mathrm e}^{\int \frac {a x^{2} \left (\tan ^{k}\left (b x \right )\right )-2}{x}d x}+\int {\mathrm e}^{\int \frac {a x^{2} \left (\tan ^{k}\left (b x \right )\right )-2}{x}d x}d x}{x y +1}\right )\]
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Added January 14, 2019.
Problem 2.6.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x -\left ( (k+1) x^k y^2- a x^{k+1} (\tan x)^m y + a(\tan x)^m \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - a*x^(k + 1)*Tan[x]^m*y + a*Tan[x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y),x)-( (k+1)*x^k*y^2- a*x^(k+1)*tan(x)^m*y + a*tan(x)^m )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {x^{k +1} {\mathrm e}^{\int \frac {a x x^{k +1} \left (\tan ^{m}x \right )-2 k -2}{x}d x}-\left (y x^{k +1}-1\right ) \left (k +1\right ) \left (\int \frac {x^{-k} {\mathrm e}^{a \left (\int x^{k +1} \left (-\frac {i \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}\right )^{m}d x \right )}}{x^{2}}d x \right )}{y x^{k +1}-1}\right )\]
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Added January 20, 2019.
Problem 2.6.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a \tan ^n(\lambda x) y^2- a b^2 \tan ^{n+2}(\lambda x) + b \lambda \tan ^2(\lambda x)+ b \lambda \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Tan[lambda*x]^n*y^2 - a*b^2*Tan[lambda*x]^(n + 2) + b*lambda*Tan[lambda*x]^2 + b*lambda)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+(a*tan(lambda*x)^n*y^2- a*b^2*tan(lambda*x)^(n+2) + b*lambda*tan(lambda*x)^2+ b*lambda)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()
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Added January 20, 2019.
Problem 2.6.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ w_x +\left ( a \tan ^k(\lambda x+\mu )(y-b x^n-c)^2 + y- b x^n + b n x^{n-1}-c \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Tan[lambda*x + mu]^k*(y - b*x^n - c)^2 + y - b*x^n + b*n*x^(n - 1) - c)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y),x)+(a *tan(lambda*x+mu)^k*(y-b*x^n-c)^2 + y- b*x^n + b*n*x^(n-1)-c)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
time expired
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Added January 20, 2019.
Problem 2.6.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ x w_x +\left ( a \tan ^m(\lambda x)y^2 +k y+ a b^2 x^{2 k} \tan ^m(\lambda x) \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = x*D[w[x, y], x] + (a*Tan[lambda*x]^m*y^2 + k*y + a*b^2*x^(2*k)*Tan[lambda*x]^m)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^xa K[1]^{k-1} \tan ^m(\lambda K[1])dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := x*diff(w(x,y),x)+ (a*tan(lambda*x)^m*y^2 +k*y+ a*b^2*x^(2*k)*tan(lambda*x)^m )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (a b \left (\int x^{k -1} \left (\tan ^{m}\left (\lambda x \right )\right )d x \right )-\arctan \left (\frac {y x^{-k}}{b}\right )\right )\]
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Added January 20, 2019.
Problem 2.6.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a \tan (\lambda x)+b) w_x +\left ( y^2+ c \tan (\mu x) y - k^2 + c k \tan (\mu x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*Tan[lambda*x] + b)*D[w[x, y], x] + (y^2 + c*Tan[mu*x]*y - k^2 + c*k*Tan[mu*x])*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*tan(lambda*x)+b)*diff(w(x,y),x)+ (y^2+ c *tan(mu*x)*y - k^2 + c*k*tan(mu*x) )*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\frac {\left (\left (k +y \right ) \left ({\mathrm e}^{2 i \mu x}+1\right )^{\frac {i a^{2} c}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \mu }} \left ({\mathrm e}^{2 i \mu x}+1\right )^{\frac {i b^{2} c}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \mu }} \left (\int \frac {\left (\frac {a \sin \left (\lambda x \right )+b \cos \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-\frac {2 a k}{\left (a^{2}+b^{2}\right ) \lambda }} \left (\frac {2}{\cos \left (2 \lambda x \right )+1}\right )^{\frac {a k}{\left (a^{2}+b^{2}\right ) \lambda }} \left ({\mathrm e}^{2 i \mu x}+1\right )^{-\frac {i a^{2} c}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \mu }} \left ({\mathrm e}^{2 i \mu x}+1\right )^{-\frac {i b^{2} c}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \mu }} \cos \left (\lambda x \right ) {\mathrm e}^{-\frac {i a^{2} c \left (\int \frac {4 i \left ({\mathrm e}^{2 i \mu x}-1\right ) a}{\left (-a +i b +\left (i b +a \right ) {\mathrm e}^{2 i \lambda x}\right ) \left ({\mathrm e}^{2 i \mu x}+1\right ) \left (i b +a \right )}d x \right )}{2 a^{2}+2 b^{2}}} {\mathrm e}^{-\frac {i b^{2} c \left (\int \frac {4 i \left ({\mathrm e}^{2 i \mu x}-1\right ) a}{\left (-a +i b +\left (i b +a \right ) {\mathrm e}^{2 i \lambda x}\right ) \left ({\mathrm e}^{2 i \mu x}+1\right ) \left (i b +a \right )}d x \right )}{2 a^{2}+2 b^{2}}} {\mathrm e}^{-\frac {2 b k \arctan \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )}{\left (a^{2}+b^{2}\right ) \lambda }} {\mathrm e}^{-\frac {a^{2} c x}{\left (a^{2}+b^{2}\right ) \left (i b +a \right )}} {\mathrm e}^{-\frac {b^{2} c x}{\left (a^{2}+b^{2}\right ) \left (i b +a \right )}}}{a \sin \left (\lambda x \right )+b \cos \left (\lambda x \right )}d x \right ) {\mathrm e}^{\frac {a^{2} c x}{\left (a^{2}+b^{2}\right ) \left (i b +a \right )}} {\mathrm e}^{\frac {b^{2} c x}{\left (a^{2}+b^{2}\right ) \left (i b +a \right )}} {\mathrm e}^{\frac {2 b k \arctan \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )}{\left (a^{2}+b^{2}\right ) \lambda }}+\left (\frac {a \sin \left (\lambda x \right )+b \cos \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )^{-\frac {2 a k}{\left (a^{2}+b^{2}\right ) \lambda }} \left (\frac {2}{\cos \left (2 \lambda x \right )+1}\right )^{\frac {a k}{\left (a^{2}+b^{2}\right ) \lambda }} {\mathrm e}^{-\frac {i a^{2} c \left (\int \frac {4 i \left ({\mathrm e}^{2 i \mu x}-1\right ) a}{\left (-a +i b +\left (i b +a \right ) {\mathrm e}^{2 i \lambda x}\right ) \left ({\mathrm e}^{2 i \mu x}+1\right ) \left (i b +a \right )}d x \right )}{2 a^{2}+2 b^{2}}} {\mathrm e}^{-\frac {i b^{2} c \left (\int \frac {4 i \left ({\mathrm e}^{2 i \mu x}-1\right ) a}{\left (-a +i b +\left (i b +a \right ) {\mathrm e}^{2 i \lambda x}\right ) \left ({\mathrm e}^{2 i \mu x}+1\right ) \left (i b +a \right )}d x \right )}{2 a^{2}+2 b^{2}}}\right ) \left ({\mathrm e}^{2 i \mu x}+1\right )^{-\frac {i a^{2} c}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \mu }} \left ({\mathrm e}^{2 i \mu x}+1\right )^{-\frac {i b^{2} c}{\left (a^{2}+b^{2}\right ) \left (i b +a \right ) \mu }} {\mathrm e}^{-\frac {2 b k \arctan \left (\frac {\sin \left (\lambda x \right )}{\cos \left (\lambda x \right )}\right )}{\left (a^{2}+b^{2}\right ) \lambda }} {\mathrm e}^{-\frac {a^{2} c x}{\left (a^{2}+b^{2}\right ) \left (i b +a \right )}} {\mathrm e}^{-\frac {b^{2} c x}{\left (a^{2}+b^{2}\right ) \left (i b +a \right )}}}{k +y}\right )\]
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Added January 20, 2019.
Problem 2.6.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n y^m + b x) w_x + \tan ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*y^m + b*x)*D[w[x, y], x] + Tan[lambda*y]^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n*y^m + b*x)*diff(w(x,y),x)+ tan(lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int y^{m} \left (\tan ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tan ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tan ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]
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Added January 20, 2019.
Problem 2.6.3.13 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b x \tan ^m y) w_x + y^k w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x*Tan[y]^m)*D[w[x, y], x] + y^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n + b*x*tan(y)^m)*diff(w(x,y),x)+ y^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int y^{-k} {\mathrm e}^{\left (n -1\right ) b \left (\int y^{-k} \left (\tan ^{m}y \right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int y^{-k} \left (\tan ^{m}y \right )d y \right )}\right )\]
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Added January 20, 2019.
Problem 2.6.3.14 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ (a x^n + b x \tan ^m y) w_x + \tan ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n + b*x*Tan[y]^m)*D[w[x, y], x] + Tan[lambda*y]^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n + b*x*tan(y)^m)*diff(w(x,y),x)+ tan(lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int \left (\tan ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tan ^{m}y \right ) \left (\tan ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tan ^{m}y \right ) \left (\tan ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]
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Added January 20, 2019.
Problem 2.6.3.15 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x^n \tan ^m y + b x) w_x + \tan ^k(\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = (a*x^n*Tan[y]^m + b*x)*D[w[x, y], x] + Tan[lambda*y]^k*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := (a*x^n*tan(y)^m+ b*x)*diff(w(x,y),x)+ tan(lambda*y)^k*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \mathit {\_F1} \left (\left (n -1\right ) a \left (\int \left (\tan ^{m}y \right ) \left (\tan ^{-k}\left (\lambda y \right )\right ) {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tan ^{-k}\left (\lambda y \right )\right )d y \right )}d y \right )+x^{-n +1} {\mathrm e}^{\left (n -1\right ) b \left (\int \left (\tan ^{-k}\left (\lambda y \right )\right )d y \right )}\right )\]
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