91 HFOPDE, chapter 3.7.2

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91.1 Problem 1

problem number 776

Added Feb. 11, 2019.

Problem Chapter 3.7.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arccos \frac {x}{\lambda }+ k \arccos \frac {y}{\beta } \]

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*ArcCos[x/lambda] + k*ArcCos[y/beta]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol[[2]] = Simplify[sol[[2]]];
 

\[ \left \{\left \{w(x,y)\to \frac {\frac {i b k x \sqrt {a^2 \left (\beta ^2-y^2\right )} \log \left (2 \left (\sqrt {a^2 \left (\beta ^2-y^2\right )}-i a y\right )\right )}{\sqrt {1-\frac {y^2}{\beta ^2}}}+a^2 b \beta c_1\left (y-\frac {b x}{a}\right )+\frac {a^2 k y^2}{\sqrt {1-\frac {y^2}{\beta ^2}}}-\frac {a^2 \beta ^2 k}{\sqrt {1-\frac {y^2}{\beta ^2}}}-\frac {i a k y \sqrt {a^2 \left (\beta ^2-y^2\right )} \log \left (2 \left (\sqrt {a^2 \left (\beta ^2-y^2\right )}-i a y\right )\right )}{\sqrt {1-\frac {y^2}{\beta ^2}}}-a b \beta c \lambda \sqrt {1-\frac {x^2}{\lambda ^2}}+a b \beta c x \cos ^{-1}\left (\frac {x}{\lambda }\right )+a b \beta k x \cos ^{-1}\left (\frac {y}{\beta }\right )}{a^2 b \beta }\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\frac {cx}{a}\arccos \left ( {\frac {x}{\lambda }} \right ) }+{\frac {ky}{b}\arccos \left ( {\frac {bx}{a\beta }}+{\frac {ya-bx}{a\beta }} \right ) }+{\frac {1}{ab} \left ( -\sqrt {-{\frac {{x}^{2}}{{\lambda }^{2}}}+1}bc\lambda -\sqrt {- \left ( {\frac {bx}{a\beta }}+{\frac {ya-bx}{a\beta }} \right ) ^{2}+1}a\beta \,k+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) ba \right ) } \]

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91.2 Problem 2

problem number 777

Added Feb. 11, 2019.

Problem Chapter 3.7.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c \arccos (\lambda x+\beta y) \]

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*D[w[x, y], y] == c*ArcCos[lambda*x + beta*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol[[2]] = Simplify[sol[[2]]];
 

\[ \left \{\left \{w(x,y)\to \frac {c \left (\beta (b x-a y) \sin ^{-1}(\beta y+\lambda x)+x (a \lambda +b \beta ) \cos ^{-1}(\beta y+\lambda x)+a \left (-\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}\right )\right )}{a (a \lambda +b \beta )}+c_1\left (y-\frac {b x}{a}\right )\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\frac {1}{a\lambda +b\beta } \left ( -\sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}c+ \left ( a\lambda +b\beta \right ) {\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) +\arccos \left ( \beta \,y+\lambda \,x \right ) c \left ( \beta \,y+\lambda \,x \right ) \right ) } \]

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91.3 Problem 3

problem number 778

Added Feb. 11, 2019.

Problem Chapter 3.7.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a x \arccos (\lambda x+\beta y) \]

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 = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*ArcCos[lambda*x + beta*y]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol[[2]] = Simplify[sol[[2]]];
 

\[ \left \{\left \{w(x,y)\to a x \left (\cos ^{-1}(\beta y+\lambda x)-\frac {\sqrt {-\beta ^2 y^2-2 \beta \lambda x y-\lambda ^2 x^2+1}}{\beta y+\lambda x}\right )+c_1\left (\frac {y}{x}\right )\right \}\right \} \]

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

\[ w \left ( x,y \right ) ={\frac {1}{\beta \,y+\lambda \,x} \left ( -\sqrt {-{\beta }^{2}{y}^{2}-2\,\beta \,\lambda \,xy-{\lambda }^{2}{x}^{2}+1}ax+ \left ( \beta \,y+\lambda \,x \right ) \left ( ax\arccos \left ( \beta \,y+\lambda \,x \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) \right ) } \]

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91.4 Problem 4

problem number 779

Added Feb. 11, 2019.

Problem Chapter 3.7.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ w \left ( x,y \right ) =\int ^{x}\! \left ( \arccos \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+{\frac {1}{a} \left ( \arccos \left ( {\frac {\beta \,\arccos \left ( \lambda \,{\it \_a} \right ) b{\it \_a}}{a}}+{\frac { \left ( -\arccos \left ( \lambda \,x \right ) bx\lambda +y\lambda \,a+\sqrt {-{\lambda }^{2}{x}^{2}+1}b \right ) \beta }{a\lambda }}-{\frac {\beta \,\sqrt {-{{\it \_a}}^{2}{\lambda }^{2}+1}b}{a\lambda }} \right ) \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {-\arccos \left ( \lambda \,x \right ) bx\lambda +y\lambda \,a+\sqrt {-{\lambda }^{2}{x}^{2}+1}b}{a\lambda }} \right ) \]

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91.5 Problem 5

problem number 780

Added Feb. 11, 2019.

Problem Chapter 3.7.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ w \left ( x,y \right ) =\int ^{y}\!{\frac {a}{b\arccos \left ( \lambda \,{\it \_a} \right ) } \left ( \pi -\arccos \left ( -{\frac {\mu \, \left ( bx\lambda +\Si \left ( \arccos \left ( y\lambda \right ) \right ) a \right ) }{\lambda \,b}}+{\frac {\mu \,\Si \left ( \arccos \left ( \lambda \,{\it \_a} \right ) \right ) a}{\lambda \,b}} \right ) \right ) ^{m}}+{\frac { \left ( \arccos \left ( \beta \,{\it \_a} \right ) \right ) ^{k}}{b\arccos \left ( \lambda \,{\it \_a} \right ) }}{d{\it \_a}}+{\it \_F1} \left ( {\frac {bx\lambda +\Si \left ( \arccos \left ( y\lambda \right ) \right ) a}{\lambda \,b}} \right ) \]