6.4.11 4.4

6.4.11.1 [1091] Problem 1

problem number 1091

Added Feb. 23, 2019.

Problem Chapter 4.4.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = (c \coth (\lambda x) + k \coth (\mu y)) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \sinh ^{\frac {c}{a \lambda }}(\lambda x) c_1\left (y-\frac {b x}{a}\right ) e^{\frac {k (\log (\tanh (\mu y))+\log (\cosh (\mu y)))}{b \mu }}\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \left (\coth \left (\lambda x \right )-1\right )^{-\frac {c}{2 a \lambda }} \left (\coth \left (\lambda x \right )+1\right )^{-\frac {c}{2 a \lambda }} \left (\coth \left (\mu y \right )-1\right )^{-\frac {k}{2 b \mu }} \left (\coth \left (\mu y \right )+1\right )^{-\frac {k}{2 b \mu }} \mathit {\_F1} \left (\frac {a y -b x}{a}\right )\]

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6.4.11.2 [1092] Problem 2

problem number 1092

Added Feb. 23, 2019.

Problem Chapter 4.4.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c \coth (\lambda x +\mu y) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == c*Coth[lambda*x + mu*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c (\log (\tanh (\lambda x+\mu y))+\log (\cosh (\lambda x+\mu y)))}{a \lambda +b \mu }\right )\right \}\right \}\]

Maple ✓

restart; 
pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) =   c*coth(lambda*x+mu*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\coth \left (\lambda x +\mu y \right )-1\right )^{-\frac {c}{2 a \lambda +2 \mu b}} \left (\coth \left (\lambda x +\mu y \right )+1\right )^{-\frac {c}{2 a \lambda +2 \mu b}} \mathit {\_F1} \left (\frac {a y -b x}{a}\right )\]

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6.4.11.3 [1093] Problem 3

problem number 1093

Added Feb. 23, 2019.

Problem Chapter 4.4.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \coth (\lambda x +\mu y) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Coth[lambda*x + mu*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) \exp \left (\frac {a x (\log (\tanh (\lambda x+\mu y))+\log (\cosh (\lambda x+\mu y)))}{\lambda x+\mu y}\right )\right \}\right \}\]

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+y*diff(w(x,y),y) =   a*x*coth(lambda*x+mu*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\coth \left (\lambda x +\mu y \right )-1\right )^{-\frac {a}{2 \left (\lambda +\frac {\mu y}{x}\right )}} \left (\coth \left (\lambda x +\mu y \right )+1\right )^{-\frac {a}{2 \left (\lambda +\frac {\mu y}{x}\right )}} \mathit {\_F1} \left (\frac {y}{x}\right )\]

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6.4.11.4 [1094] Problem 4

problem number 1094

Added Feb. 23, 2019.

Problem Chapter 4.4.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b \coth ^n(\lambda x) w_y = (c \coth ^m(\mu x)+s \coth ^k(\beta y)) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b \coth ^{n+1}(\lambda x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\lambda x)\right )}{a \lambda n+a \lambda }\right ) \exp \left (\int _1^x\frac {s \coth ^k\left (\frac {\beta \left (-b \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\lambda x)\right ) \coth ^{n+1}(\lambda x)+a \lambda (n+1) y+b \coth ^{n+1}(\lambda K[1]) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(\lambda K[1])\right )\right )}{a \lambda (n+1)}\right )+c \coth ^m(\mu K[1])}{a}dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (y -\left (\int \frac {b \left (\coth ^{n}\left (\lambda x \right )\right )}{a}d x \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {c \left (\coth ^{m}\left (\mathit {\_b} \mu \right )\right )+s \left (-\coth \left (\left (-y -\left (\int \frac {b \left (\coth ^{n}\left (\mathit {\_b} \lambda \right )\right )}{a}d \mathit {\_b} \right )+\int \frac {b \left (\coth ^{n}\left (\lambda x \right )\right )}{a}d x \right ) \beta \right )\right )^{k}}{a}d\mathit {\_b}}\]

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6.4.11.5 [1095] Problem 5

problem number 1095

Added Feb. 23, 2019.

Problem Chapter 4.4.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b \coth ^n(\lambda y) w_y = (c \coth ^m(\mu x)+s \coth ^k(\beta y)) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\coth ^{1-n}(\lambda y) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\coth ^2(\lambda y)\right )}{\lambda -\lambda n}-\frac {b x}{a}\right ) \exp \left (\int _1^y\frac {\left (s \coth ^k(\beta K[1])+c \coth ^m\left (\frac {-a \mu \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\coth ^2(\lambda y)\right ) \coth ^{1-n}(\lambda y)+b \lambda \mu x-b \lambda \mu n x+a \mu \coth ^{1-n}(\lambda K[1]) \, _2F_1\left (1,\frac {1}{2}-\frac {n}{2};\frac {3}{2}-\frac {n}{2};\coth ^2(\lambda K[1])\right )}{b \lambda -b \lambda n}\right )\right ) \coth ^{-n}(\lambda K[1])}{b}dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\frac {a \left (\int \left (\coth ^{-n}\left (\lambda y \right )\right )d y \right )}{b}+x \right ) {\mathrm e}^{\int _{}^{y}\frac {\left (c \left (-\coth \left (-\mu \left (\int \frac {a \left (\coth ^{-n}\left (\mathit {\_b} \lambda \right )\right )}{b}d \mathit {\_b} \right )-\left (-\frac {a \left (\int \left (\coth ^{-n}\left (\lambda y \right )\right )d y \right )}{b}+x \right ) \mu \right )\right )^{m}+s \left (\coth ^{k}\left (\mathit {\_b} \beta \right )\right )\right ) \left (\coth ^{-n}\left (\mathit {\_b} \lambda \right )\right )}{b}d\mathit {\_b}}\]

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