Added May 31, 2019.
Problem Chapter 6.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + (c \sin ^n(\lambda x)+s \cos ^k(\beta y) ) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +(c*Sin[lambda*x]^n+s*Cos[beta*y]^k)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\frac {c \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{a \lambda n+a \lambda }+\frac {s \sqrt {\sin ^2(\beta y)} \csc (\beta y) \cos ^{k+1}(\beta y) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\beta y)\right )}{b \beta k+b \beta }+z\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+(c*sin(lambda*x)^n+s*cos(beta*y)^k)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},-\int ^{x}\!{\frac {1}{a} \left ( c \left ( \sin \left ( {\it \_a}\,\lambda \right ) \right ) ^{n}+s \left ( \cos \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{k} \right ) }{d{\it \_a}}+z \right ) \]
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Added May 31, 2019.
Problem Chapter 6.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \sin (\beta y) w_y + c \cos (\lambda x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] +c*Cos[lambda*x]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tan \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},z-\frac {c \sin (\lambda x)}{a \lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*sin(beta*y)*diff(w(x,y,z),y)+c*cos(lambda*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{\beta \,b}\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( 2\,{{\it \_Z}{{\rm e}^{{\frac {b\beta \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}},-{ \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) },{\frac {za\lambda -c\sin \left ( x\lambda \right ) }{a\lambda }} \right ) \]
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Added May 31, 2019.
Problem Chapter 6.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \cos ^k(\beta x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] +b*Cos[beta*x]^k*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{k+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\beta x)\right )}{\beta k+\beta }+z,y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ a*sin(lambda*x)^n*diff(w(x,y,z),y)+b*cos(beta*x)^k*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \sin \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cos \left ( \beta \,x \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]
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Added May 31, 2019.
Problem Chapter 6.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \sin ^n(\lambda x) w_y + b \sin ^k(\beta y) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] +b*Sin[beta*y]^k*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y,z),x)+ a*sin(lambda*x)^n*diff(w(x,y,z),y)+b*sin(beta*y)^k*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \sin \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \sin \left ( \beta \, \left ( a\int \! \left ( \sin \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}-\int \!a \left ( \sin \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{k}{d{\it \_b}}+z \right ) \]
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Added May 31, 2019.
Problem Chapter 6.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \tan (\beta y) w_y + c \cot (\lambda x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Tan[beta*y]*D[w[x, y,z], y] +c*Cot[lambda*x]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (z-\frac {c \log (\sin (\lambda x))}{a \lambda },\frac {\log (\sin (\beta y))}{\beta }-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*tan(beta*y)*diff(w(x,y,z),y)+c*cot(lambda*x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {1}{\beta \,b} \left ( -b\beta \,x+\ln \left ( {\tan \left ( \beta \,y \right ) {\frac {1}{\sqrt {1+ \left ( \tan \left ( \beta \,y \right ) \right ) ^{2}}}}} \right ) a \right ) },{\frac {2\,za\lambda +c\ln \left ( \left ( \cot \left ( x\lambda \right ) \right ) ^{2}+1 \right ) }{2\,a\lambda }} \right ) \]
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Added May 31, 2019.
Problem Chapter 6.6.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + a \cot ^n(\lambda x) w_y + b \tan ^k(\beta y) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cot[lambda*x]^n*D[w[x, y,z], y] +b*Tan[beta*y]^k*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y,z),x)+ a*cot(lambda*x)^n*diff(w(x,y,z),y)+b*tan(beta*y)^k*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \cot \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( {\frac {-\tan \left ( \beta \, \left ( -\int \!a \left ( \cot \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) -\tan \left ( \beta \,a\int \! \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) }{\tan \left ( \beta \, \left ( -\int \!a \left ( \cot \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \tan \left ( \beta \,a\int \! \left ( \cot \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) -1}} \right ) ^{k}{d{\it \_b}}+z \right ) \]
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