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'));
 

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

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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'));
 

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

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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'));
 

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

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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]];
 

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

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'));
 

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

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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]];
 

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

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'));
 

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

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