Added January 20, 2019.
Problem 2.6.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \cot ^k(\lambda x)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Cot[lambda*x]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {a \cot ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};-\cot ^2(\lambda x)\right )}{k \lambda +\lambda }-b x+y\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*cot(lambda*x)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -bx-\int \!a \left ( \cot \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+y \right ) \] Has unresolved integral
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Added January 20, 2019.
Problem 2.6.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( a \cot ^k(\lambda y)+b \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (a*Cot[lambda*y]^k + b)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\frac {1}{a \cot ^k(\lambda K[1])+b}dK[1]-x\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ (a*cot(lambda*y)^k+b)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( a \left ( \cot \left ( \lambda \,y \right ) \right ) ^{k}+b \right ) ^{-1}\,{\rm d}y+x \right ) \]
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Added January 20, 2019.
Problem 2.6.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + a \cot ^k(x+\lambda y) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + a*Cot[x + 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 := diff(w(x,y),x)+ cot(x+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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -\lambda \,\int ^{{\frac {\lambda \,y+x}{\lambda }}}\! \left ( 1+ \left ( \cot \left ( \lambda \,{\it \_a} \right ) \right ) ^{k}\lambda \right ) ^{-1}{d{\it \_a}}+x \right ) \]
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Added January 20, 2019.
Problem 2.6.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+a \lambda + a(\lambda -a) \cot ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + a*lambda + a*(lambda - a)*Cot[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)*cot(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( \left ( -\cos \left ( \lambda \,x \right ) a+a\cos \left ( 3\,\lambda \,x \right ) +3\,y\sin \left ( \lambda \,x \right ) -y\sin \left ( 3\,\lambda \,x \right ) \right ) \LegendreP \left ( {\frac {2\,a-\lambda }{2\,\lambda }},{\frac {2\,a-\lambda }{2\,\lambda }},\cos \left ( \lambda \,x \right ) \right ) -2\,\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \lambda \, \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) \right ) \left ( \left ( \cos \left ( \lambda \,x \right ) a-a\cos \left ( 3\,\lambda \,x \right ) -3\,y\sin \left ( \lambda \,x \right ) +y\sin \left ( 3\,\lambda \,x \right ) \right ) \LegendreQ \left ( {\frac {2\,a-\lambda }{2\,\lambda }},{\frac {2\,a-\lambda }{2\,\lambda }},\cos \left ( \lambda \,x \right ) \right ) +2\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) \lambda \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2+\lambda ^2 + 3 a \lambda +a(\lambda -a) \cot ^2(\lambda x) \right ) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 + lambda^2 + 3*a*lambda + a*(lambda - a)*Cot[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)*cot(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( { \left ( 4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\LegendreP \left ( 1/2\,{\frac {2\,a+3\,\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \lambda -2\,\LegendreP \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \left ( 1/2\,\lambda \, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}+y \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sin \left ( \lambda \,x \right ) + \left ( \left ( a+3/2\,\lambda \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-\lambda /2 \right ) \cos \left ( \lambda \,x \right ) -y\sin \left ( \lambda \,x \right ) \right ) \right ) \left ( -4\, \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}\LegendreQ \left ( 1/2\,{\frac {2\,a+3\,\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \lambda +2\,\LegendreQ \left ( 1/2\,{\frac {2\,a+\lambda }{\lambda }},1/2\,{\frac {2\,a-\lambda }{\lambda }},\cos \left ( \lambda \,x \right ) \right ) \left ( 1/2\,\lambda \, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{3}+y \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}\sin \left ( \lambda \,x \right ) + \left ( \left ( a+3/2\,\lambda \right ) \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}-\lambda /2 \right ) \cos \left ( \lambda \,x \right ) -y\sin \left ( \lambda \,x \right ) \right ) \right ) ^{-1}} \right ) \]
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Added January 20, 2019.
Problem 2.6.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ w_x + \left ( y^2-2 a \cot (a x) y + b^2-a^2 \right ) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y], x] + (y^2 - 2*a*Cot[a*x]*y + b^2 - a^2)*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\tan ^{-1}\left (\frac {y-a \cot (a x)}{\sqrt {b^2}}\right )-\sqrt {b^2} x\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y),x)+ ( y^2-2*a*cot(a*x)*y + b^2-a^2)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {{{\rm e}^{-2\,ibx}} \left ( i\cot \left ( ax \right ) a-iy-b \right ) }{2\,b \left ( ib-a\cot \left ( ax \right ) +y \right ) }} \right ) \]
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Added January 20, 2019.
Problem 2.6.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \cot (\lambda x) w_x + a \cot (\mu y) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = Cot[lambda*x]*D[w[x, y], x] + a*Cot[mu*y]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \cos (\mu y) \cos ^{-\frac {a \mu }{\lambda }}(\lambda x)}{\mu }\right )\right \}\right \}\]
Maple ✓
restart; pde := cot(lambda*x)*diff(w(x,y),x)+ a*cot(mu*y)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{2\,\mu \,\lambda } \left ( \ln \left ( {\frac { \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1}{ \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}}} \right ) a\mu +\lambda \,\ln \left ( \left ( \cos \left ( \mu \,y \right ) \right ) ^{2} \right ) \right ) } \right ) \]
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Added January 20, 2019.
Problem 2.6.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \cot (\mu y) w_x + a \cot (\lambda x) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = Cot[mu*y]*D[w[x, y], x] + a*Cot[lambda*x]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \sin (\mu y) \sin ^{-\frac {a \mu }{\lambda }}(\lambda x)}{\mu }\right )\right \}\right \}\]
Maple ✓
restart; pde := cot(mu*y)*diff(w(x,y),x)+ a*cot(lambda*x)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a\mu }\ln \left ( {\frac {\tan \left ( \mu \,y \right ) }{ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1}\sqrt { \left ( \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \left ( -2\, \left ( -1+\cos \left ( 2\,\lambda \,x \right ) \right ) ^{-1} \right ) ^{{\frac {a\mu }{\lambda }}}}} \right ) } \right ) \]
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Added January 20, 2019.
Problem 2.6.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \cot (\mu y) w_x + a \cot ^2(\lambda x) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = Cot[mu*y]*D[w[x, y], x] + a*Cot[lambda*x]^2*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {4 \sin (\mu y) e^{\frac {a \mu (\lambda x+\cot (\lambda x))}{\lambda }}}{\mu }\right )\right \}\right \}\]
Maple ✓
restart; pde := cot(mu*y)*diff(w(x,y),x)+ a*cot(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'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{2\,\sin \left ( \lambda \,x \right ) a\lambda \,\mu } \left ( \ln \left ( {\frac { \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}}{ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1}} \right ) \lambda \,\sin \left ( \lambda \,x \right ) +2\,a \left ( {\rm arccot} \left ({\frac {\cos \left ( \lambda \,x \right ) }{\sin \left ( \lambda \,x \right ) }}\right )\sin \left ( \lambda \,x \right ) -1/2\,\pi \,\sin \left ( \lambda \,x \right ) +\cos \left ( \lambda \,x \right ) \right ) \mu \right ) } \right ) \]
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Added January 20, 2019.
Problem 2.6.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \cot (y+a) w_x + c \cot (x+b) w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = Cot[y + a]*D[w[x, y], x] + c*Cot[x + b]^2*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (4 \sin (a+y) e^{c (\cot (b+x)+b+x)}\right )\right \}\right \}\]
Maple ✓
restart; pde := cot(y+a)*diff(w(x,y),x)+ c*cot(x+b)^2*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{2\,\tan \left ( b \right ) \left ( \tan \left ( x \right ) +\tan \left ( b \right ) \right ) } \left ( -2\,\tan \left ( b \right ) \left ( \tan \left ( x \right ) +\tan \left ( b \right ) \right ) \ln \left ( {\frac {\cos \left ( y \right ) \tan \left ( a \right ) +\sin \left ( y \right ) }{\sin \left ( y \right ) \tan \left ( a \right ) }} \right ) +\tan \left ( b \right ) \left ( \tan \left ( x \right ) +\tan \left ( b \right ) \right ) \ln \left ( \left ( \sin \left ( y \right ) \right ) ^{-2} \right ) +2\, \left ( \left ( \tan \left ( x \right ) -x+\pi /2 \right ) \left ( \tan \left ( b \right ) \right ) ^{2}- \left ( x-\pi /2 \right ) \tan \left ( x \right ) \tan \left ( b \right ) +\tan \left ( x \right ) \right ) c \right ) } \right ) \]
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Added January 20, 2019.
Problem 2.6.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ \cot (\lambda x) \cot (\mu y) w_x + a w_y = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = Cot[lambda*x]*Cot[mu*y]*D[w[x, y], x] + a*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c_1\left (\frac {2 \sin (\mu y) \cos ^{\frac {a \mu }{\lambda }}(\lambda x)}{\mu }\right )\right \}\right \}\]
Maple ✓
restart; pde := cot(lambda*x)*cot(mu*y)*diff(w(x,y),x)+ a*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a\mu }\ln \left ( {\frac {\tan \left ( \mu \,y \right ) }{ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1}\sqrt { \left ( \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}+1 \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2\,{\frac {a\mu }{\lambda }}}}} \right ) } \right ) \]
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Added January 20, 2019.
Problem 2.6.4.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\) \[ \cot (\lambda x) \cot (\mu y) w_x + a \cot (v x) w_y = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = Cot[lambda*x]*Cot[mu*y]*D[w[x, y], x] + a*Cot[v*x]*D[w[x, y], y] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := cot(lambda*x)*cot(mu*y)*diff(w(x,y),x)+ a*cot(v*x)*diff(w(x,y),y) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {1}{a\mu }\ln \left ( {{\it csgn} \left ( \left ( \cos \left ( \mu \,y \right ) \right ) ^{-1} \right ) \sin \left ( \mu \,y \right ) {{\rm e}^{a\mu \,x}} \left ( {{\rm e}^{2\,ivx}}-1 \right ) ^{{\frac {ia\mu }{v}}} \left ( {{\rm e}^{a\mu \,\int \!{\frac {-2\,{{\rm e}^{2\,ivx}}-2}{ \left ( {{\rm e}^{2\,ivx}}-1 \right ) \left ( {{\rm e}^{2\,i\lambda \,x}}+1 \right ) }}\,{\rm d}x}} \right ) ^{-1}} \right ) } \right ) \]
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