6.8.17 6.4

6.8.17.1 [1859] Problem 1

problem number 1859

Added Oct 18, 2019.

Problem Chapter 8.6.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a w_y + b w_z = c \cot ^n(\beta x) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x) \exp \left (-\frac {c \cot ^{n+1}(\beta x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(\beta x)\right )}{\beta n+\beta }\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-a x +y , -b x +z \right ) {\mathrm e}^{\int c \left (\cot ^{n}\left (\beta x \right )\right )d x}\]

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6.8.17.2 [1860] Problem 2

problem number 1860

Added Oct 18, 2019.

Problem Chapter 8.6.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a w_x + b w_y + c \cot (\beta z) w_z = \left ( k \cot (\lambda x)+s \cot (\gamma y) \right ) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +  c*Cot[beta*z]*D[w[x,y,z],z]== (k*Cot[lambda*x]+s*Cot[gamma*y])*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

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

Maple ✓

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

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

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6.8.17.3 [1861] Problem 3

problem number 1861

Added Oct 18, 2019.

Problem Chapter 8.6.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ w_x + a \cot ^n(\beta x) w_y + b \cot ^k(\lambda x) w_z = c \cot ^m(\gamma x) w \]

Mathematica ✓

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

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

Maple ✓

restart; 
local gamma; 
pde :=  diff(w(x,y,z),x)+a*cot(beta*x)^n*diff(w(x,y,z),y)+ b*cot(lambda*x)^k*diff(w(x,y,z),z)= c*cot(gamma*x)^m*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int a \left (\cot ^{n}\left (\beta x \right )\right )d x \right ), z -\left (\int b \left (\cot ^{k}\left (\lambda x \right )\right )d x \right )\right ) {\mathrm e}^{\int c \left (\cot ^{m}\left (\gamma x \right )\right )d x}\]

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6.8.17.4 [1862] Problem 4

problem number 1862

Added Oct 18, 2019.

Problem Chapter 8.6.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a w_x + b \cot (\beta y) w_y + c \cot (\lambda x) w_z = k \cot (\gamma z) w \]

Mathematica ✓

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

\begin {align*} & \left \{w(x,y,z)\to c_1\left (\frac {c \cot ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\lambda x)\right )+a \lambda m z+a \lambda z}{a \lambda m+a \lambda },\frac {\log (\sec (\beta y))}{\beta }-\frac {b x}{a}\right ) \exp \left (\int _1^x\frac {k \cot \left (\frac {\gamma \left (c \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\lambda x)\right ) \cot ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cot ^{m+1}(\lambda K[1]) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\lambda K[1])\right )\right )}{a \lambda (m+1)}\right )}{a}dK[1]\right )\right \}\\& \left \{w(x,y,z)\to c_1\left (\frac {c \cot ^{m+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\lambda x)\right )+a \lambda m z+a \lambda z}{a \lambda m+a \lambda },\frac {\log (\sec (\beta y))}{\beta }-\frac {b x}{a}\right ) \exp \left (\int _1^x\frac {k \cot \left (\frac {\gamma \left (c \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\lambda x)\right ) \cot ^{m+1}(\lambda x)+a \lambda (m+1) z-c \cot ^{m+1}(\lambda K[2]) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\cot ^2(\lambda K[2])\right )\right )}{a \lambda (m+1)}\right )}{a}dK[2]\right )\right \}\\ \end {align*}

Maple ✓

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

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )}{2 b \beta }, z -\left (\int _{}^{y}\frac {c \left (\frac {\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right ) \cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right ) \cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )-\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right )+\cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right )+\cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )}{\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right ) \cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )+\left (\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right )-\cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )\right ) \cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right )+1}\right )^{m}}{b \cot \left (\mathit {\_a} \beta \right )}d\mathit {\_a} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {k \cot \left (\left (z +\int \frac {c \left (\frac {\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_b} \beta \right )\right )}{b \beta }\right ) \cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right ) \cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_b} \beta \right )+1\right )}{2 b \beta }\right )-\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_b} \beta \right )\right )}{b \beta }\right )+\cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right )+\cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_b} \beta \right )+1\right )}{2 b \beta }\right )}{\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_b} \beta \right )\right )}{b \beta }\right ) \cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_b} \beta \right )+1\right )}{2 b \beta }\right )+\left (\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_b} \beta \right )\right )}{b \beta }\right )-\cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_b} \beta \right )+1\right )}{2 b \beta }\right )\right ) \cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right )+1}\right )^{m}}{b \cot \left (\mathit {\_b} \beta \right )}d \mathit {\_b} -\left (\int _{}^{y}\frac {c \left (\frac {\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right ) \cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right ) \cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )-\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right )+\cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right )+\cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )}{\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right ) \cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )+\left (\cot \left (\frac {a \lambda \ln \left (\cot \left (\mathit {\_a} \beta \right )\right )}{b \beta }\right )-\cot \left (\frac {a \lambda \ln \left (\cot ^{2}\left (\mathit {\_a} \beta \right )+1\right )}{2 b \beta }\right )\right ) \cot \left (\frac {\left (2 b \beta x -a \ln \left (\cot ^{2}\left (\beta y \right )+1\right )+2 a \ln \left (\cot \left (\beta y \right )\right )\right ) \lambda }{2 b \beta }\right )+1}\right )^{m}}{b \cot \left (\mathit {\_a} \beta \right )}d\mathit {\_a} \right )\right ) \gamma \right )}{b \cot \left (\mathit {\_b} \beta \right )}d\mathit {\_b}}\]

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6.8.17.5 [1863] Problem 5

problem number 1863

Added Oct 18, 2019.

Problem Chapter 8.6.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y,z)\) \[ a_1 \cot ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cot ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cot ^{n_2}(\lambda _2 x) + b_2 \cot ^{m_2}(\beta _2 y) + c_2 \cot ^{k_2}(\gamma _2 z) \right ) w \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a1*Cot[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cot[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Cot[lambda2*z]^n2 + b2*Cot[beta2*y]^m2 + c2*Cot[gamma2*z]^k2)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde :=  a1*cot(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cot(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*cot(lambda2*z)^n2 + b2*cot(beta2*y)^m2 + c2*cot(gamma2*z)^k2)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z , x -\left (\int _{}^{y}\frac {\mathit {a1} \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right ) \left (\cot ^{\mathit {n1}}\left (\lambda 1 \RootOf \left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\mathit {\_b} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_b} \right )\right )\right )\right )}{\mathit {b1}}d\mathit {\_f} \right )\right ) {\mathrm e}^{\int _{}^{y}\frac {\left (\mathit {a2} \left (\cot ^{\mathit {n2}}\left (\lambda 2 \RootOf \left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right )+\mathit {b2} \left (\cot ^{\mathit {m2}}\left (\mathit {\_f} \beta 2 \right )\right )+\mathit {c2} \left (\cot ^{\mathit {k2}}\left (\gamma 2 \RootOf \left (\int \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )d \mathit {\_f} -\left (\int \left (\cot ^{-\mathit {m1}}\left (\beta 1 y \right )\right )d y \right )+\int \frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\gamma 1 z \right )\right )}{\mathit {c1}}d z -\left (\int ^{\mathit {\_Z}}\frac {\mathit {b1} \left (\cot ^{-\mathit {k1}}\left (\mathit {\_a} \gamma 1 \right )\right )}{\mathit {c1}}d \mathit {\_a} \right )\right )\right )\right )\right ) \left (\cot ^{-\mathit {m1}}\left (\mathit {\_f} \beta 1 \right )\right )}{\mathit {b1}}d\mathit {\_f}}\]

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