7.6.18 6.5

7.6.18.1 [1526] Problem 1

problem number 1526

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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (\frac {a y -b x}{a}, z -\left (\int _{}^{x}\frac {c \left (\sin ^{n}\left (\textit {\_a} \lambda \right )\right )+s \left (\cos ^{k}\left (\frac {\left (a y -\left (-\textit {\_a} +x \right ) b \right ) \beta }{a}\right )\right )}{a}d \textit {\_a} \right )\right )\]

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7.6.18.2 [1527] Problem 2

problem number 1527

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'));
 

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

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7.6.18.3 [1528] Problem 3

problem number 1528

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]];
 

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

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int b \left (\cos ^{k}\left (\beta x \right )\right )d x \right )\right )\]

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7.6.18.4 [1529] Problem 4

problem number 1529

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int _{}^{x}b \left (\sin ^{k}\left (\left (a \left (\int \left (\sin ^{n}\left (\textit {\_b} \lambda \right )\right )d \textit {\_b} \right )+y -\left (\int a \left (\sin ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )\right )d \textit {\_b} \right )\right )\]

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7.6.18.5 [1530] Problem 5

problem number 1530

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'));
 

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

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7.6.18.6 [1531] Problem 6

problem number 1531

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'));
 

\[w \left (x , y , z\right ) = \textit {\_F1} \left (y -\left (\int a \left (\cot ^{n}\left (\lambda x \right )\right )d x \right ), z -\left (\int _{}^{x}b \left (\frac {-\tan \left (\left (y -\left (\int a \left (\cot ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right )-\tan \left (a \beta \left (\int \left (\cot ^{n}\left (\textit {\_b} \lambda \right )\right )d \textit {\_b} \right )\right )}{\tan \left (\left (y -\left (\int a \left (\cot ^{n}\left (\lambda x \right )\right )d x \right )\right ) \beta \right ) \tan \left (a \beta \left (\int \left (\cot ^{n}\left (\textit {\_b} \lambda \right )\right )d \textit {\_b} \right )\right )-1}\right )^{k}d \textit {\_b} \right )\right )\]

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