93 HFOPDE, chapter 3.7.4

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93.1 Problem 1

problem number 786

Added Feb. 11, 2019.

Problem Chapter 3.7.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

\[ w \left ( x,y \right ) =1/2\,{\frac {x}{ab} \left ( \pi \,bc+\pi \,bk-2\,\arctan \left ( {\frac {x}{\lambda }} \right ) bc \right ) }-{\frac {ky}{b}\arctan \left ( {\frac {bx}{a\beta }}+{\frac {ya-bx}{a\beta }} \right ) }+1/2\,{\frac {1}{ab} \left ( k\beta \,\ln \left ( \left ( {\frac {bx}{a\beta }}+{\frac {ya-bx}{a\beta }} \right ) ^{2}+1 \right ) a+\lambda \,c\ln \left ( {\frac {{x}^{2}}{{\lambda }^{2}}}+1 \right ) b+2\,{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) ba \right ) } \]

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93.2 Problem 2

problem number 787

Added Feb. 11, 2019.

Problem Chapter 3.7.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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93.3 Problem 3

problem number 788

Added Feb. 11, 2019.

Problem Chapter 3.7.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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

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

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93.4 Problem 4

problem number 789

Added Feb. 11, 2019.

Problem Chapter 3.7.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

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

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93.5 Problem 5

problem number 790

Added Feb. 11, 2019.

Problem Chapter 3.7.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[ w \left ( x,y \right ) =\int ^{y}\!{\frac {a}{b{\rm arccot} \left ({\it \_b}\,\lambda \right )} \left ( \pi /2-\arctan \left ( 2\,{\frac {\mu \,a\int \! \left ( \pi -2\,\arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-1}\,{\rm d}{\it \_b}}{b}}+\mu \, \left ( -\int \!2\,{\frac {a}{b \left ( \pi -2\,\arctan \left ( y\lambda \right ) \right ) }}\,{\rm d}y+x \right ) \right ) \right ) ^{m}}+{\frac { \left ( \pi /2-\arctan \left ( \beta \,{\it \_b} \right ) \right ) ^{k}}{b{\rm arccot} \left ({\it \_b}\,\lambda \right )}}{d{\it \_b}}+{\it \_F1} \left ( -\int \!2\,{\frac {a}{b \left ( \pi -2\,\arctan \left ( y\lambda \right ) \right ) }}\,{\rm d}y+x \right ) \]