6.4.10 4.3

6.4.10.1 [1086] Problem 1

problem number 1086

Added Feb. 23, 2019.

Problem Chapter 4.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.10.2 [1087] Problem 2

problem number 1087

Added Feb. 23, 2019.

Problem Chapter 4.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.10.3 [1088] Problem 3

problem number 1088

Added Feb. 23, 2019.

Problem Chapter 4.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.10.4 [1089] Problem 4

problem number 1089

Added Feb. 23, 2019.

Problem Chapter 4.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.10.5 [1090] Problem 5

problem number 1090

Added Feb. 23, 2019.

Problem Chapter 4.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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