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'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a} \left ( {\it c1}\, \left ( \sinh \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+{\it c2}\, \left ( \cosh \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n} \right ) {{\rm e}^{-{\frac {{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {x}{a}}}}\]

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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'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac { \left ( \sinh \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}}{a} \left ( \cosh \left ( {\frac {\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) }{a}} \right ) \right ) ^{n}{{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]

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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'));
 

\[w \left ( x,y \right ) ={{\rm e}^{{\frac {cx}{a}}}} \left ( {\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) +\int ^{x}\!-{\frac {1}{a} \left ( \left ( k-s \right ) \cosh \left ( {\frac { \left ( -{\it \_a}\,\lambda +\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) -\cosh \left ( {\frac { \left ( {\it \_a}\,\lambda +\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) \left ( k+s \right ) \right ) {{\rm e}^{-{\frac {c{\it \_a}}{a}}}} \left ( \sinh \left ( {\frac { \left ( {\it \_a}\,\lambda +\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) +\sinh \left ( {\frac { \left ( -{\it \_a}\,\lambda +\mu \,y \right ) a-b\mu \, \left ( x-{\it \_a} \right ) }{a}} \right ) \right ) ^{-1}}{d{\it \_a}} \right ) \]

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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'));
 

\[w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {k}{a}\cosh \left ( {\frac {\mu \, \left ( ya\lambda -b\cosh \left ( x\lambda \right ) +b\cosh \left ( {\it \_a}\,\lambda \right ) \right ) }{a\lambda }} \right ) {{\rm e}^{-{\frac {c{\it \_a}}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya\lambda -b\cosh \left ( x\lambda \right ) }{a\lambda }} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}}\]

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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'));
 

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

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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}{a} \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n} \left ( {\frac {\cosh \left ( \beta \,{\it \_f} \right ) }{\sinh \left ( \beta \,{\it \_f} \right ) }} \right ) ^{s}{{\rm e}^{-{\frac {c}{a}\int \! \left ( {\sinh \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\cosh \left ( \mu \,{\it \_f} \right ) }{\sinh \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ay-b\int \! \left ( {\frac {\cosh \left ( \mu \,x \right ) }{\sinh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \left ( \cosh \left ( {\frac {\nu }{a} \left ( b\int \! \left ( {\frac {\cosh \left ( \mu \,{\it \_f} \right ) }{\sinh \left ( \mu \,{\it \_f} \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}+ay-b\int \! \left ( {\frac {\cosh \left ( \mu \,x \right ) }{\sinh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) ^{-1}} \right ) ^{k} \left ( {\frac {\sinh \left ( {\it \_f}\,\lambda \right ) }{\cosh \left ( {\it \_f}\,\lambda \right ) }} \right ) ^{-n}\,{\rm d}{\it \_f}}}}}{d{\it \_f}}+{\it \_F1} \left ( {\frac {1}{a} \left ( ay-b\int \! \left ( {\frac {\cosh \left ( \mu \,x \right ) }{\sinh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac { \left ( \tanh \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}c}{a} \left ( {\sinh \left ( {\frac {\nu }{a} \left ( \int \!{\frac {b}{a} \left ( {\frac {\cosh \left ( \mu \,{\it \_b} \right ) }{\sinh \left ( \mu \,{\it \_b} \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( {\it \_b}\,\lambda \right ) }{\cosh \left ( {\it \_b}\,\lambda \right ) }} \right ) ^{-n}}\,{\rm d}{\it \_b}a+ay-b\int \! \left ( {\frac {\cosh \left ( \mu \,x \right ) }{\sinh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \left ( \cosh \left ( {\frac {\nu }{a} \left ( \int \!{\frac {b}{a} \left ( {\frac {\cosh \left ( \mu \,{\it \_b} \right ) }{\sinh \left ( \mu \,{\it \_b} \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( {\it \_b}\,\lambda \right ) }{\cosh \left ( {\it \_b}\,\lambda \right ) }} \right ) ^{-n}}\,{\rm d}{\it \_b}a+ay-b\int \! \left ( {\frac {\cosh \left ( \mu \,x \right ) }{\sinh \left ( \mu \,x \right ) }} \right ) ^{m} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-n}\,{\rm d}x \right ) } \right ) \right ) ^{-1}} \right ) ^{k}}{d{\it \_b}}}}\]

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