6.6.14 6.1

6.6.14.1 [1504] Problem 1

problem number 1504

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},-\frac {c x}{a}-\frac {\text {arctanh}(\cos (\gamma z))}{\gamma }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {x c \gamma +a \ln \left (\csc \left (\gamma z \right )+\cot \left (\gamma z \right )\right )}{c \gamma }, \frac {y c \gamma +b \ln \left (\csc \left (\gamma z \right )+\cot \left (\gamma z \right )\right )}{c \gamma }\right )\]

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6.6.14.2 [1505] Problem 2

problem number 1505

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\beta y))}{\beta },z-\int _1^x\frac {c \sin (\lambda K[1])}{a}dK[1]\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {x b \beta +a \ln \left (\csc \left (\beta y \right )+\cot \left (\beta y \right )\right )}{b \beta }, \frac {z \lambda a +c \cos \left (\lambda x \right )}{\lambda a}\right )\]

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6.6.14.3 [1506] Problem 3

problem number 1506

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (-\frac {b x}{a}-\frac {\text {arctanh}(\cos (\beta y))}{\beta },-\frac {c x}{a}-\frac {\text {arctanh}(\cos (\gamma z))}{\gamma }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y , z\right )=f_{1} \left (x \right ) f_{2} \left (y \right ) f_{3} \left (z \right )\boldsymbol {\operatorname {where}}\left [\left \{\frac {d}{d x}f_{1} \left (x \right )=\textit {\_c}_{1} f_{1} \left (x \right ), \frac {d}{d y}f_{2} \left (y \right )=\textit {\_c}_{2} f_{2} \left (y \right ) \csc \left (\beta y \right ), \frac {d}{d z}f_{3} \left (z \right )=-\frac {f_{3} \left (z \right ) \left (a \textit {\_c}_{1}+b \textit {\_c}_{2}\right ) \csc \left (\gamma z \right )}{c}\right \}\right ]\]

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6.6.14.4 [1507] Problem 4

problem number 1507

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]];
 

Failed

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {\ln \left (\csc \left (\beta y \right )+\cot \left (\beta y \right )\right ) a \lambda -\cos \left (\lambda x \right ) b \beta }{\beta a \lambda }, \frac {z b \beta \sqrt {\frac {a^{2} \lambda ^{2}}{b^{2} \beta ^{2}}}+c \,\operatorname {csgn}\left (\sin \left (\lambda x \right )\right ) \arctan \left (\frac {\sqrt {\frac {a^{2} \lambda ^{2}}{b^{2} \beta ^{2}}}\, b \beta \cot \left (\lambda x \right )}{a \lambda }\right )}{\sqrt {\frac {a^{2} \lambda ^{2}}{b^{2} \beta ^{2}}}\, b \beta }\right )\]

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6.6.14.5 [1508] Problem 5

problem number 1508

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (-\int \sin \left (\lambda x \right )^{n}d x +\frac {a \int \sin \left (\beta y \right )^{-m}d y}{b}, -\int \sin \left (\mu x \right )^{k}d x +\frac {a \int \sin \left (\gamma z \right )^{-r}d z}{c}\right )\]

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