7.6.15 6.2

7.6.15.1 [1511] Problem 1

problem number 1511

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, \frac {a \ln \left (\RootOf \left (\gamma z -\arctan \left (\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}+1}, \frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {\gamma c x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}+1}\right )\right )\right )}{\gamma c}\right )\]

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7.6.15.2 [1512] Problem 2

problem number 1512

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\beta y -\arctan \left (\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}, \frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {b \beta x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}\right )\right )\right )}{b \beta }, \frac {a \lambda z -c \sin \left (\lambda x \right )}{a \lambda }\right )\]

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7.6.15.3 [1513] Problem 3

problem number 1513

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\beta y -\arctan \left (\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}, \frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {b \beta x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}\right )\right )\right )}{b \beta }, \frac {a \ln \left (\RootOf \left (\gamma z -\arctan \left (\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}+1}, \frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {\gamma c x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}+1}\right )\right )\right )}{\gamma c}\right )\]

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7.6.15.4 [1514] Problem 4

problem number 1514

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a \ln \left (\RootOf \left (\beta y -\arctan \left (\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}, \frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {b \beta x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 b \beta x}{a}}+1}\right )\right )\right )}{b \beta }, \frac {a \ln \left (\RootOf \left (\gamma z -\arctan \left (\frac {\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}-1}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}+1}, \frac {2 \textit {\_Z} \,{\mathrm e}^{\frac {\gamma c x}{a}}}{\textit {\_Z}^{2} {\mathrm e}^{\frac {2 \gamma c x}{a}}+1}\right )\right )\right )}{\gamma c}\right )\]

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7.6.15.5 [1515] Problem 5

problem number 1515

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {b \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right )}{a \lambda n+a \lambda }+\frac {\sqrt {\sin ^2(\beta y)} \csc (\beta y) \cos ^{1-m}(\beta y) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-m}{2},\frac {3-m}{2},\cos ^2(\beta y)\right )}{\beta (m-1)},\frac {c \sqrt {\sin ^2(\mu x)} \csc (\mu x) \cos ^{k+1}(\mu x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\cos ^2(\mu x)\right )}{a k \mu +a \mu }+\frac {\sqrt {\sin ^2(\gamma z)} \csc (\gamma z) \cos ^{1-r}(\gamma z) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-r}{2},\frac {3-r}{2},\cos ^2(\gamma z)\right )}{\gamma (r-1)}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (-\left (\int \left (\cos ^{n}\left (\lambda x \right )\right )d x \right )+\int \frac {a \left (\cos ^{-m}\left (\beta y \right )\right )}{b}d y , -\left (\int \left (\cos ^{k}\left (\mu x \right )\right )d x \right )+\int \frac {a \left (\cos ^{-r}\left (\gamma z \right )\right )}{c}d z \right )\]

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