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'));
 

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

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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'));
 

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

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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'));
 

\[w \left (x , y\right ) = \frac {a \ln \left (\cosh \left (\lambda x +\mu y \right )\right ) x}{\lambda x +\mu y}+f_{1} \left (\frac {y}{x}\right )\]

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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]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {s \tanh ^k\left (\frac {\beta \left (-b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\lambda x)\right ) \tanh ^{n+1}(\lambda x)+b \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\lambda K[1])\right ) \tanh ^{n+1}(\lambda K[1])+a \lambda (n+1) y\right )}{a \lambda (n+1)}\right )+c \tanh ^m(\mu K[1])}{a}dK[1]+c_1\left (y-\frac {b \tanh ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},\tanh ^2(\lambda x)\right )}{a \lambda n+a \lambda }\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {b \int \tanh \left (\lambda x \right )^{n}d x}{a}+y \right )+\frac {\int _{}^{x}\left ({\tanh \left (\frac {\beta \left (\int \tanh \left (\lambda \textit {\_b} \right )^{n}d \textit {\_b} b -\int \tanh \left (\lambda x \right )^{n}d x b +y a \right )}{a}\right )}^{k} s +c \tanh \left (\mu \textit {\_b} \right )^{m}\right )d \textit {\_b}}{a}\]

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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]];
 

\[\left \{\left \{w(x,y)\to \int _1^y\frac {\tanh ^{-n}(\lambda K[1]) \left (s \tanh ^k(\beta K[1])+c \tanh ^m\left (\frac {-a \mu \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\tanh ^2(\lambda y)\right ) \tanh ^{1-n}(\lambda y)+a \mu \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\tanh ^2(\lambda K[1])\right ) \tanh ^{1-n}(\lambda K[1])+b \lambda \mu x-b \lambda \mu n x}{b \lambda -b \lambda n}\right )\right )}{b}dK[1]+c_1\left (\frac {\tanh ^{1-n}(\lambda y) \operatorname {Hypergeometric2F1}\left (1,\frac {1-n}{2},\frac {3-n}{2},\tanh ^2(\lambda y)\right )}{\lambda -\lambda n}-\frac {b x}{a}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = f_{1} \left (-\frac {a \int \tanh \left (\lambda y \right )^{-n}d y}{b}+x \right )+\frac {\int _{}^{y}\tanh \left (\lambda \textit {\_b} \right )^{-n} \left ({\tanh \left (\frac {\mu \left (a \int \tanh \left (\lambda \textit {\_b} \right )^{-n}d \textit {\_b} -a \int \tanh \left (\lambda y \right )^{-n}d y +x b \right )}{b}\right )}^{m} c +s \tanh \left (\beta \textit {\_b} \right )^{k}\right )d \textit {\_b}}{b}\]

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