6.6.18 6.5

6.6.18.1 [1524] Problem 1

problem number 1524

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'));
 

\[w \left (x , y , z\right ) = f_{1} \left (\frac {a y -x b}{a}, -\frac {\int _{}^{x}\left (c \sin \left (\lambda \textit {\_a} \right )^{n}+s \cos \left (\frac {\beta \left (a y -b \left (x -\textit {\_a} \right )\right )}{a}\right )^{k}\right )d \textit {\_a}}{a}+z \right )\]

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6.6.18.2 [1525] Problem 2

problem number 1525

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]];
 

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

Maple ✓

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

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

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6.6.18.3 [1526] Problem 3

problem number 1526

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'));
 

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

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6.6.18.4 [1527] Problem 4

problem number 1527

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda },z-\int _1^xb \sin ^k\left (\frac {\beta \left (-a \sqrt {\cos ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right ) \sec (\lambda x) \sin ^{n+1}(\lambda x)+a \sqrt {\cos ^2(\lambda K[1])} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda K[1])\right ) \sec (\lambda K[1]) \sin ^{n+1}(\lambda K[1])+\lambda (n+1) y\right )}{\lambda (n+1)}\right )dK[1]\right )\right \}\right \}\]

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'));
 

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

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6.6.18.5 [1528] Problem 5

problem number 1528

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'));
 

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

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6.6.18.6 [1529] Problem 6

problem number 1529

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \cot ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right )}{\lambda n+\lambda }+y,z-\int _1^xb \tan ^k\left (\frac {\beta \left (a \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda x)\right ) \cot ^{n+1}(\lambda x)+\lambda (n+1) y-a \cot ^{n+1}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\lambda K[1])\right )\right )}{\lambda (n+1)}\right )dK[1]\right )\right \}\right \}\]

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'));
 

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

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