78 HFOPDE, chapter 3.4.2

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78.1 Problem 1

problem number 713

Added Feb. 9, 2019.

Problem Chapter 3.4.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cosh[lambda*x] + k*Cosh[mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

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

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78.2 Problem 2

problem number 714

Added Feb. 9, 2019.

Problem Chapter 3.4.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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*Cosh[lambda*x + mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

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

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78.3 Problem 3

problem number 715

Added Feb. 9, 2019.

Problem Chapter 3.4.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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] == a*x*Cosh[lambda*x + mu*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

\[ w \left ( x,y \right ) ={\frac {x}{ \left ( a\lambda +b\mu \right ) ^{2}} \left ( \sinh \left ( {\frac { \left ( a\lambda +b\mu \right ) x}{a}}+{\frac { \left ( ya-bx \right ) \mu }{a}} \right ) {a}^{2}\lambda +\sinh \left ( {\frac { \left ( a\lambda +b\mu \right ) x}{a}}+{\frac { \left ( ya-bx \right ) \mu }{a}} \right ) ab\mu \right ) }+{\frac {1}{ \left ( a\lambda +b\mu \right ) ^{2}} \left ( {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {a}^{2}{\lambda }^{2}+2\,{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) {b}^{2}{\mu }^{2}-\cosh \left ( {\frac { \left ( a\lambda +b\mu \right ) x}{a}}+{\frac { \left ( ya-bx \right ) \mu }{a}} \right ) {a}^{2} \right ) } \]

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78.4 Problem 4

problem number 716

Added Feb. 9, 2019.

Problem Chapter 3.4.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cosh ^n(\lambda x) w_y = c \cosh ^m(\mu x)+ s \cosh ^k(\beta y) \]

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*Cosh[lambda*x]^n*D[w[x, y], y] == c*Cosh[mu*x]^m + s*Cosh[beta*y]^k; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \]

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

\[ w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( c \left ( \cosh \left ( {\it \_b}\,\mu \right ) \right ) ^{m}+s \left ( \cosh \left ( {\frac {\beta }{a} \left ( \int \! \left ( \cosh \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}\,{\rm d}{\it \_b}b+ \left ( -\int \!{\frac {b \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) a \right ) } \right ) \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!{\frac {b \left ( \cosh \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \]

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78.5 Problem 5

problem number 717

Added Feb. 9, 2019.

Problem Chapter 3.4.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \cosh ^n(\lambda y) w_y = c \cosh ^m(\mu x)+ s \cosh ^k(\beta y) \]

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*Cosh[lambda*y]^n*D[w[x, y], y] == c*Cosh[mu*x]^m + s*Cosh[beta*y]^k; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[ \text {\$Aborted} \] Timed out

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

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