7.6.14 6.1

7.6.14.1 [1506] Problem 1

problem number 1506

Added May 26, 2019.

Problem Chapter 6.6.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b w_y + c \sin (\gamma z) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*Sin[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\log \left (\tan \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+c*sin(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'));
 

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

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7.6.14.2 [1507] Problem 2

problem number 1507

Added May 26, 2019.

Problem Chapter 6.6.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \sin (\beta y) w_y + c \sin (\lambda x) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] +c*Sin[lambda*x]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {c \cos (\lambda x)}{a \lambda }+z,\frac {\log \left (\tan \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*sin(beta*y)*diff(w(x,y,z),y)+c*sin(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'));
 

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

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7.6.14.3 [1508] Problem 3

problem number 1508

Added May 26, 2019.

Problem Chapter 6.6.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \sin (\beta y) w_y + c \sin (\gamma z) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Sin[beta*y]*D[w[x, y,z], y] +c*Sin[gamma*z]*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tan \left (\frac {\beta y}{2}\right )\right )}{\beta }-\frac {b x}{a},\frac {\log \left (\tan \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*sin(beta*y)*diff(w(x,y,z),y)+c*sin(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'));
 

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

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7.6.14.4 [1509] Problem 4

problem number 1509

Added May 26, 2019.

Problem Chapter 6.6.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \sin (\lambda x) \sin (\beta y) w_y + c w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Sin[lambda*x]*Sin[beta*y]*D[w[x, y,z], y] +c*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (z-\frac {c x}{a},\frac {a \log \left (\tan ^2\left (\frac {\beta y}{2}\right )\right )}{\beta }+\frac {2 b \cos (\lambda x)}{\lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+ b*sin(lambda*x)*sin(beta*y)*diff(w(x,y,z),y)+c*diff(w(x,y,z),z)= 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

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

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7.6.14.5 [1510] Problem 5

problem number 1510

Added May 26, 2019.

Problem Chapter 6.6.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\)

\[ a w_x + b \sin ^n(\lambda x) \sin ^m(\beta y) w_y + c \sin ^k(\mu x) \sin ^r(\gamma *z) w_z = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Sin[lambda*x]^n*Sin[beta*y]^m*D[w[x, y,z], y] +c*Sin[mu*x]^k*Sin[gamma*z]^r*D[w[x,y,z],z]==0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  a*diff(w(x,y,z),x)+  b*sin(lambda*x)^n*sin(beta*y)^m*diff(w(x,y,z),y)+c*sin(mu*x)^k*sin(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'));
 

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

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