6.5.9 4.3

6.5.9.1 [1260] Problem 1

problem number 1260

Added April 3, 2019.

Problem Chapter 5.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.9.2 [1261] Problem 2

problem number 1261

Added April 3, 2019.

Problem Chapter 5.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.9.3 [1262] Problem 3

problem number 1262

Added April 3, 2019.

Problem Chapter 5.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = \left (c_1 \tanh ^{n_1}(\lambda _1 x)+ c_2 \tanh ^{n_2}(\lambda _2 y) \right ) w + s_1 \tanh ^{k_1}(\beta _1 x)+ s_2 \tanh ^{k_2}(\beta _2 y) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == (c1*Tanh[lambda1*x]^n1 + c2*Tanh[lambda2*y]^n2)*w[x,y] + s1*Tanh[beta1*x]^k1+ s2*Tanh[beta2*y]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$Aborted

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c1*tanh(lambda1*x)^n1 + c2*tanh(lambda2*y)^n2)*w(x,y) + s1*tanh(beta1*x)^k1+ s2*tanh(beta2*y)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {s1} \left (\tanh ^{\mathit {k1}}\left (\mathit {\_b} \beta 1 \right )\right )+\mathit {s2} \left (\tanh ^{\mathit {k2}}\left (\frac {\left (a y -\left (-\mathit {\_b} +x \right ) b \right ) \beta 2}{a}\right )\right )\right ) {\mathrm e}^{-\frac {\int \left (\mathit {c1} \left (\tanh ^{\mathit {n1}}\left (\mathit {\_b} \lambda 1 \right )\right )+\mathit {c2} \left (\tanh ^{\mathit {n2}}\left (\frac {\left (a y -\left (-\mathit {\_b} +x \right ) b \right ) \lambda 2}{a}\right )\right )\right )d \mathit {\_b}}{a}}}{a}d\mathit {\_b} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {c1} \left (\tanh ^{\mathit {n1}}\left (\mathit {\_a} \lambda 1 \right )\right )+\mathit {c2} \left (\tanh ^{\mathit {n2}}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \lambda 2}{a}\right )\right )}{a}d\mathit {\_a}}\]

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6.5.9.4 [1263] Problem 4

problem number 1263

Added April 3, 2019.

Problem Chapter 5.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.9.5 [1264] Problem 5

problem number 1264

Added April 3, 2019.

Problem Chapter 5.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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