Added May 26, 2019.
Problem Chapter 6.6.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b w_y + c \cos (\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*Cos[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\begin {align*} & \left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\frac {\cosh ^{-1}\left (-\frac {\sec (\gamma z) \left (2 \left (2 \sec (\gamma z) \sqrt {\sin ^2(\gamma z) \cos ^2(\gamma z) \sinh ^2\left (\frac {c \gamma x}{a}\right ) \left (\cosh \left (\frac {4 c \gamma x}{a}\right )-\sinh \left (\frac {4 c \gamma x}{a}\right )\right )}+\sinh ^3\left (\frac {c \gamma x}{a}\right )+\sinh \left (\frac {c \gamma x}{a}\right )\right )-2 \cosh ^3\left (\frac {c \gamma x}{a}\right )+\left (1-3 \cosh \left (\frac {2 c \gamma x}{a}\right )\right ) \cosh \left (\frac {c \gamma x}{a}\right )+6 \sinh \left (\frac {c \gamma x}{a}\right ) \cosh ^2\left (\frac {c \gamma x}{a}\right )\right )}{4 \cosh \left (\frac {2 c \gamma x}{a}\right )-4 \sinh \left (\frac {2 c \gamma x}{a}\right )}\right )}{\gamma }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\cosh ^{-1}\left (-\frac {\sec (\gamma z) \left (2 \left (2 \sec (\gamma z) \sqrt {\sin ^2(\gamma z) \cos ^2(\gamma z) \sinh ^2\left (\frac {c \gamma x}{a}\right ) \left (\cosh \left (\frac {4 c \gamma x}{a}\right )-\sinh \left (\frac {4 c \gamma x}{a}\right )\right )}+\sinh ^3\left (\frac {c \gamma x}{a}\right )+\sinh \left (\frac {c \gamma x}{a}\right )\right )-2 \cosh ^3\left (\frac {c \gamma x}{a}\right )+\left (1-3 \cosh \left (\frac {2 c \gamma x}{a}\right )\right ) \cosh \left (\frac {c \gamma x}{a}\right )+6 \sinh \left (\frac {c \gamma x}{a}\right ) \cosh ^2\left (\frac {c \gamma x}{a}\right )\right )}{4 \cosh \left (\frac {2 c \gamma x}{a}\right )-4 \sinh \left (\frac {2 c \gamma x}{a}\right )}\right )}{\gamma }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\frac {\cosh ^{-1}\left (\frac {\sec (\gamma z) \left (-2 \left (-2 \sec (\gamma z) \sqrt {\sin ^2(\gamma z) \cos ^2(\gamma z) \sinh ^2\left (\frac {c \gamma x}{a}\right ) \left (\cosh \left (\frac {4 c \gamma x}{a}\right )-\sinh \left (\frac {4 c \gamma x}{a}\right )\right )}+\sinh ^3\left (\frac {c \gamma x}{a}\right )+\sinh \left (\frac {c \gamma x}{a}\right )\right )+2 \cosh ^3\left (\frac {c \gamma x}{a}\right )+\left (3 \cosh \left (\frac {2 c \gamma x}{a}\right )-1\right ) \cosh \left (\frac {c \gamma x}{a}\right )-6 \sinh \left (\frac {c \gamma x}{a}\right ) \cosh ^2\left (\frac {c \gamma x}{a}\right )\right )}{4 \cosh \left (\frac {2 c \gamma x}{a}\right )-4 \sinh \left (\frac {2 c \gamma x}{a}\right )}\right )}{\gamma }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\cosh ^{-1}\left (\frac {\sec (\gamma z) \left (-2 \left (-2 \sec (\gamma z) \sqrt {\sin ^2(\gamma z) \cos ^2(\gamma z) \sinh ^2\left (\frac {c \gamma x}{a}\right ) \left (\cosh \left (\frac {4 c \gamma x}{a}\right )-\sinh \left (\frac {4 c \gamma x}{a}\right )\right )}+\sinh ^3\left (\frac {c \gamma x}{a}\right )+\sinh \left (\frac {c \gamma x}{a}\right )\right )+2 \cosh ^3\left (\frac {c \gamma x}{a}\right )+\left (3 \cosh \left (\frac {2 c \gamma x}{a}\right )-1\right ) \cosh \left (\frac {c \gamma x}{a}\right )-6 \sinh \left (\frac {c \gamma x}{a}\right ) \cosh ^2\left (\frac {c \gamma x}{a}\right )\right )}{4 \cosh \left (\frac {2 c \gamma x}{a}\right )-4 \sinh \left (\frac {2 c \gamma x}{a}\right )}\right )}{\gamma }\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*cos(gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {ay-xb}{a}},{\frac {a}{c\gamma }\ln \left ( \RootOf \left ( \gamma \,z-\arctan \left ( { \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {c\gamma \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cos (\beta y) w_y + c \cos (\lambda x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cos[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]];
\begin {align*} & \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },z-\frac {c \sin (\lambda x)}{a \lambda }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {\cosh ^{-1}\left (-\frac {\sec (\beta y) \left (2 \left (2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )-2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (1-3 \cosh \left (\frac {2 b \beta x}{a}\right )\right ) \cosh \left (\frac {b \beta x}{a}\right )+6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },z-\frac {c \sin (\lambda x)}{a \lambda }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (-\frac {\cosh ^{-1}\left (\frac {\sec (\beta y) \left (-2 \left (-2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )+2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (3 \cosh \left (\frac {2 b \beta x}{a}\right )-1\right ) \cosh \left (\frac {b \beta x}{a}\right )-6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },z-\frac {c \sin (\lambda x)}{a \lambda }\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {\cosh ^{-1}\left (\frac {\sec (\beta y) \left (-2 \left (-2 \sec (\beta y) \sqrt {\sin ^2(\beta y) \cos ^2(\beta y) \sinh ^2\left (\frac {b \beta x}{a}\right ) \left (\cosh \left (\frac {4 b \beta x}{a}\right )-\sinh \left (\frac {4 b \beta x}{a}\right )\right )}+\sinh ^3\left (\frac {b \beta x}{a}\right )+\sinh \left (\frac {b \beta x}{a}\right )\right )+2 \cosh ^3\left (\frac {b \beta x}{a}\right )+\left (3 \cosh \left (\frac {2 b \beta x}{a}\right )-1\right ) \cosh \left (\frac {b \beta x}{a}\right )-6 \sinh \left (\frac {b \beta x}{a}\right ) \cosh ^2\left (\frac {b \beta x}{a}\right )\right )}{4 \cosh \left (\frac {2 b \beta x}{a}\right )-4 \sinh \left (\frac {2 b \beta x}{a}\right )}\right )}{\beta },z-\frac {c \sin (\lambda x)}{a \lambda }\right )\right \}\\ \end {align*}
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*cos(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'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{\beta \,b}\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( { \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {b\beta \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) },{\frac {za\lambda -c\sin \left ( x\lambda \right ) }{a\lambda }} \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cos (\beta y) w_y + c \cos (\gamma z) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cos[beta*y]*D[w[x, y,z], y] +c*Cos[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*cos(beta*y)*diff(w(x,y,z),y)+c*cos(gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{\beta \,b}\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( { \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {b\beta \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) },{\frac {a}{c\gamma }\ln \left ( \RootOf \left ( \gamma \,z-\arctan \left ( { \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {c\gamma \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cos (\beta y) w_y + c \cos (\gamma z) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cos[beta*y]*D[w[x, y,z], y] +c*Cos[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
$Aborted
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*cos(beta*y)*diff(w(x,y,z),y)+c*cos(gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {a}{\beta \,b}\ln \left ( \RootOf \left ( \beta \,y-\arctan \left ( { \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {b\beta \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {b\beta \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) },{\frac {a}{c\gamma }\ln \left ( \RootOf \left ( \gamma \,z-\arctan \left ( { \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}-1 \right ) \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}},2\,{{\it \_Z}{{\rm e}^{{\frac {c\gamma \,x}{a}}}} \left ( {{\it \_Z}}^{2}{{\rm e}^{2\,{\frac {c\gamma \,x}{a}}}}+1 \right ) ^{-1}} \right ) \right ) \right ) } \right ) \]
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Added May 26, 2019.
Problem Chapter 6.6.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ a w_x + b \cos ^n(\lambda x) \cos ^m(\beta y) w_y + c \cos ^k(\mu x) \cos ^r(\gamma *z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = a*D[w[x, y,z], x] + b*Cos[lambda*x]^n*Cos[beta*y]^m*D[w[x, y,z], y] +c*Cos[mu*x]^k*Cos[gamma*z]^r*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(\lambda x)\right )}{a \lambda n+a \lambda }+\frac {\sqrt {\sin ^2(\beta y)} \csc (\beta y) \cos ^{1-m}(\beta y) \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(\beta y)\right )}{\beta (m-1)},\frac {c \sqrt {\sin ^2(\mu x)} \csc (\mu x) \cos ^{k+1}(\mu x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\cos ^2(\mu x)\right )}{a k \mu +a \mu }+\frac {\sqrt {\sin ^2(\gamma z)} \csc (\gamma z) \cos ^{1-r}(\gamma z) \, _2F_1\left (\frac {1}{2},\frac {1-r}{2};\frac {3-r}{2};\cos ^2(\gamma z)\right )}{\gamma (r-1)}\right )\right \}\right \}\]
Maple ✓
restart; pde := a*diff(w(x,y,z),x)+ b*cos(lambda*x)^n*cos(beta*y)^m*diff(w(x,y,z),y)+c*cos(mu*x)^k*cos(gamma*z)^r*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \cos \left ( x\lambda \right ) \right ) ^{n}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \beta \,y \right ) \right ) ^{-m}a}{b}}\,{\rm d}y,-\int \! \left ( \cos \left ( \mu \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \gamma \,z \right ) \right ) ^{-r}a}{c}}\,{\rm d}z \right ) \]
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