81 HFOPDE, chapter 3.4.5

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81.1 Problem 1

problem number 728

Added Feb. 11, 2019.

Problem Chapter 3.4.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

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

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*Sinh[lambda*x] + k*Cosh[mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {a b \lambda \mu c_1\left (\frac {a y-b x}{a}\right )+a k \lambda \sinh \left (\frac {b \mu x}{a}\right ) \cosh \left (\frac {\mu (a y-b x)}{a}\right )+a k \lambda \cosh \left (\frac {b \mu x}{a}\right ) \sinh \left (\frac {\mu (a y-b x)}{a}\right )+b c \mu \cosh (\lambda x)}{a b \lambda \mu }\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) =  c*sinh(lambda*x)+ k*cosh(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {1}{b\mu \,a\lambda } \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +\cosh \left ( \lambda \,x \right ) cb\mu +k\sinh \left ( {\frac { \left ( ya-bx \right ) \mu }{a}}+{\frac {b\mu \,x}{a}} \right ) a\lambda \right ) } \]

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81.2 Problem 2

problem number 729

Added Feb. 11, 2019.

Problem Chapter 3.4.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \tanh (\lambda x) + k \coth (\mu y) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == Tanh[lambda*x] + k*Coth[mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {a b \lambda \mu c_1\left (\frac {a y-b x}{a}\right )+a k \lambda \log \left (\sinh \left (\frac {\mu (a y-b x)}{a}+\frac {b \mu x}{a}\right )\right )+b \mu \log (\cosh (\lambda x))}{a b \lambda \mu }\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) =  tanh(lambda*x)+ k*coth(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac { \left ( bk\lambda \,\mu -b\lambda \,\mu \right ) x}{b\mu \,a\lambda }}-2\,{\frac {ky}{b}}+{\frac {1}{b\mu \,a\lambda } \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +k\ln \left ( {{\rm e}^{2\,{\frac { \left ( ya-bx \right ) \mu }{a}}+2\,{\frac {b\mu \,x}{a}}}}-1 \right ) a\lambda +\ln \left ( {{\rm e}^{2\,\lambda \,x}}+1 \right ) b\mu \right ) } \]

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81.3 Problem 3

problem number 730

Added Feb. 11, 2019.

Problem Chapter 3.4.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b w_y = \sinh (\lambda x) + k \tanh (\mu y) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == Sinh[lambda*x] + k*Tanh[mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {a b \lambda \mu c_1\left (\frac {a y-b x}{a}\right )+a k \lambda \log \left (\cosh \left (\frac {\mu (a y-b x)}{a}+\frac {b \mu x}{a}\right )\right )+b \mu \cosh (\lambda x)}{a b \lambda \mu }\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
pde :=a*diff(w(x,y),x) + b*diff(w(x,y),y) =  sinh(lambda*x)+ k*tanh(mu*y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {kx}{a}}-2\,{\frac {ky}{b}}+1/2\,{\frac {1}{b\mu \,a\lambda } \left ( 2\,{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +2\,k\ln \left ( {{\rm e}^{2\,{\frac { \left ( ya-bx \right ) \mu }{a}}+2\,{\frac {b\mu \,x}{a}}}}+1 \right ) a\lambda +{{\rm e}^{\lambda \,x}}b\mu +{{\rm e}^{-\lambda \,x}}b\mu \right ) } \]

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81.4 Problem 4

problem number 731

Added Feb. 11, 2019.

Problem Chapter 3.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \cosh (\mu y)w_y = \sinh (\lambda x) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 pde = a*D[w[x, y], x] + b*Cosh[mu*y]*D[w[x, y], y] == Sinh[lambda*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {a \lambda c_1\left (\frac {2 a \tan ^{-1}\left (\tanh \left (\frac {\mu y}{2}\right )\right )-b \mu x}{a \mu }\right )+\cosh (\lambda x)}{a \lambda }\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
pde :=a*diff(w(x,y),x) + b*cosh(mu*y)*diff(w(x,y),y) =  sinh(lambda*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( {\frac {-b\mu \,x+2\,\arctan \left ( {{\rm e}^{\mu \,y}} \right ) a}{b\mu }} \right ) a\lambda +\cosh \left ( \lambda \,x \right ) \right ) } \]

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81.5 Problem 5

problem number 732

Added Feb. 11, 2019.

Problem Chapter 3.4.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a w_x + b \sinh (\mu y)w_y = \cosh (\lambda x) \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s, lambda, B, s, mu, d, g, B, v, f, h, q, p, delta]; 
 pde = a*D[w[x, y], x] + b*Sinh[mu*y]*D[w[x, y], y] == Cosh[lambda*x]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{w(x,y)\to \frac {a \lambda c_1\left (\frac {a \log \left (\tanh \left (\frac {\mu y}{2}\right )\right )-b \mu x}{a \mu }\right )+\sinh (\lambda x)}{a \lambda }\right \}\right \} \]

Maple ✓

 
w:='w';x:='x';y:='y';a:='a';b:='b';n:='n';m:='m';c:='c';k:='k';alpha:='alpha';beta:='beta';g:='g';A:='A';f:='f'; 
C:='C';lambda:='lambda';B:='B';mu:='mu';d:='d';s:='s';v:='v';q:='q';p:='p';l:='l'; 
pde :=a*diff(w(x,y),x) + b*sinh(mu*y)*diff(w(x,y),y) =  cosh(lambda*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac {1}{a\lambda } \left ( {\it \_F1} \left ( -{\frac {b\mu \,x+2\,\arctanh \left ( {{\rm e}^{\mu \,y}} \right ) a}{b\mu }} \right ) a\lambda +\sinh \left ( \lambda \,x \right ) \right ) } \]