6.6.15 6.2

6.6.15.1 [1509] Problem 1

problem number 1509

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {b x}{a},\frac {\coth ^{-1}(\sin (\gamma z))}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

Maple ✓

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

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

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6.6.15.2 [1510] Problem 2

problem number 1510

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]];
 

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

Maple ✓

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

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

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6.6.15.3 [1511] Problem 3

problem number 1511

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\coth ^{-1}(\sin (\beta y))}{\beta }-\frac {b x}{a},\frac {\coth ^{-1}(\sin (\gamma z))}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

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'));
 

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

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6.6.15.4 [1512] Problem 4

problem number 1512

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]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\coth ^{-1}(\sin (\beta y))}{\beta }-\frac {b x}{a},\frac {\coth ^{-1}(\sin (\gamma z))}{\gamma }-\frac {c x}{a}\right )\right \}\right \}\]

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'));
 

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

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6.6.15.5 [1513] Problem 5

problem number 1513

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'));
 

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

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