89 HFOPDE, chapter 3.6.5

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89.1 Problem 1

problem number 765

Added Feb. 11, 2019.

Problem Chapter 3.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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89.2 Problem 2

problem number 766

Added Feb. 11, 2019.

Problem Chapter 3.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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] == Tan[lambda*x] + c*Sin[mu*y] + k; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol[[2]] = Simplify[sol[[2]]];
 

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

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

\[ w \left ( x,y \right ) ={\frac {1}{b\mu \,a\lambda } \left ( -2\,c \left ( \cos \left ( 1/2\,\mu \,y \right ) \right ) ^{2}a\lambda +{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) b\mu \,a\lambda +kxb\mu \,\lambda -\ln \left ( {\frac {\sin \left ( 1/2\,\lambda \,x \right ) -\cos \left ( 1/2\,\lambda \,x \right ) }{\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) b\mu -\ln \left ( {\frac {\sin \left ( 1/2\,\lambda \,x \right ) +\cos \left ( 1/2\,\lambda \,x \right ) }{\cos \left ( 1/2\,\lambda \,x \right ) }} \right ) b\mu +\ln \left ( \left ( \cos \left ( 1/2\,\lambda \,x \right ) \right ) ^{-2} \right ) b\mu \right ) } \]

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89.3 Problem 3

problem number 767

Added Feb. 11, 2019.

Problem Chapter 3.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ \left \{\left \{w(x,y)\to \frac {2 a^3 \lambda ^2 c_1\left (\frac {a y-b x}{a}\right )-a^2 \lambda \cos \left (\mu \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )+\lambda x\right )+2 a^2 c \lambda ^2 x-a^2 \lambda \cos (\lambda x-\mu y)-2 a b^2 \mu ^2 c_1\left (\frac {a y-b x}{a}\right )-a b \mu \cos (\lambda x-\mu y)+a b \mu \cos \left (\mu \left (\frac {a y-b x}{a}+\frac {b x}{a}\right )+\lambda x\right )-2 b^2 c \mu ^2 x}{2 a (a \lambda -b \mu ) (a \lambda +b \mu )}\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\frac {cx}{a}}+{\frac {1}{a} \left ( -1/2\,{\frac {a}{a\lambda -b\mu }\cos \left ( {\frac { \left ( a\lambda -b\mu \right ) x}{a}}-{\frac { \left ( ya-bx \right ) \mu }{a}} \right ) }-1/2\,{\frac {a}{a\lambda +b\mu }\cos \left ( {\frac { \left ( a\lambda +b\mu \right ) x}{a}}+{\frac { \left ( ya-bx \right ) \mu }{a}} \right ) } \right ) }+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \]

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89.4 Problem 4

problem number 768

Added Feb. 11, 2019.

Problem Chapter 3.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \sin (\mu y) w_y = \cos (\lambda y)+c \]

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

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

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

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

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89.5 Problem 5

problem number 769

Added Feb. 11, 2019.

Problem Chapter 3.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tan (\mu y) w_y = \sin (\lambda y)+c \]

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

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

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

\[ w \left ( x,y \right ) ={\frac {cx}{a}}-{\frac {1}{a\lambda } \left ( -{\it \_F1} \left ( {\frac {1}{b\mu } \left ( -b\mu \,x+\ln \left ( {\frac {\tan \left ( \mu \,y \right ) }{\sqrt {1+ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}}}} \right ) a \right ) } \right ) a\lambda +\cos \left ( \lambda \,x \right ) \right ) } \]

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89.6 Problem 6

problem number 770

Added Feb. 11, 2019.

Problem Chapter 3.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tan (\mu y) w_y = \cot (\lambda y)+c \]

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

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

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

\[ w \left ( x,y \right ) ={\frac {cx}{a}}+1/2\,{\frac {1}{a\lambda } \left ( 2\,{\it \_F1} \left ( {\frac {1}{b\mu } \left ( -b\mu \,x+\ln \left ( {\frac {\tan \left ( \mu \,y \right ) }{\sqrt {1+ \left ( \tan \left ( \mu \,y \right ) \right ) ^{2}}}} \right ) a \right ) } \right ) a\lambda -\ln \left ( \left ( \cot \left ( \lambda \,x \right ) \right ) ^{2}+1 \right ) \right ) } \]