6.3.10 4.3

6.3.10.1 [909] Problem 1

problem number 909

Added Feb. 9, 2019.

Problem Chapter 3.4.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.10.2 [910] Problem 2

problem number 910

Added Feb. 9, 2019.

Problem Chapter 3.4.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.10.3 [911] Problem 3

problem number 911

Added Feb. 11, 2019.

Problem Chapter 3.4.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + y w_y = a x \tanh (\lambda x+\mu y) \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.10.4 [912] Problem 4

problem number 912

Added Feb. 11, 2019.

Problem Chapter 3.4.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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6.3.10.5 [913] Problem 5

problem number 913

Added Feb. 11, 2019.

Problem Chapter 3.4.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

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; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

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

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