6.5.11 4.5

6.5.11.1 [1270] Problem 1

problem number 1270

Added April 4, 2019.

Problem Chapter 5.4.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = w + c_1 \sinh ^k(\lambda x) +c_2 \cosh ^n(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*Sinh[lambda*x]^k+c2*Cosh[beta*y]^n; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} c_1\left (y-\frac {b x}{a}\right )+\frac {\text {c1} \left (e^{2 \lambda x}-1\right ) \sinh ^k(\lambda x) \, _2F_1\left (1,\frac {1}{2} \left (k-\frac {1}{a \lambda }+2\right );\frac {1}{2} \left (-k-\frac {1}{a \lambda }+2\right );e^{2 \lambda x}\right )}{a k \lambda +1}-\frac {\text {c2} \left (e^{2 \beta y}+1\right ) \cosh ^n(\beta y) \, _2F_1\left (1,\frac {1}{2} \left (n-\frac {1}{b \beta }+2\right );\frac {1}{2} \left (-n-\frac {1}{b \beta }+2\right );-e^{2 \beta y}\right )}{b \beta n+1}\right \}\right \}\]

Maple ✓

restart; 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*sinh(lambda*x)^k+c2*cosh(beta*y)^n; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {\left (\mathit {c1} \left (\sinh ^{k}\left (\mathit {\_a} \lambda \right )\right )+\mathit {c2} \left (\cosh ^{n}\left (\frac {\left (a y -\left (-\mathit {\_a} +x \right ) b \right ) \beta }{a}\right )\right )\right ) {\mathrm e}^{-\frac {\mathit {\_a}}{a}}}{a}d\mathit {\_a} +\mathit {\_F1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]

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6.5.11.2 [1271] Problem 2

problem number 1271

Added April 4, 2019.

Problem Chapter 5.4.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.11.3 [1272] Problem 3

problem number 1272

Added April 4, 2019.

Problem Chapter 5.4.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.11.4 [1273] Problem 4

problem number 1273

Added April 4, 2019.

Problem Chapter 5.4.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.11.5 [1274] Problem 5

problem number 1274

Added April 4, 2019.

Problem Chapter 5.4.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.5.11.6 [1275] Problem 6

problem number 1275

Added April 4, 2019.

Problem Chapter 5.4.5.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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