____________________________________________________________________________________
Added March 9, 2019.
Problem Chapter 4.7.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c \arccot (\frac {x}{\lambda } + k \arccot (\frac {y}{\beta } ) \right ) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == (c*ArcCot[x/lambda] + k*ArcCot[y/beta])*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to \left (\lambda ^2+x^2\right )^{\frac {c \lambda }{2 a}} c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {k \left (a \beta \log \left (a^2 \left (\beta ^2+y^2\right )\right )+2 \tan ^{-1}\left (\frac {y}{\beta }\right ) (b x-a y)+2 b x \cot ^{-1}\left (\frac {y}{\beta }\right )\right )+2 b c x \cot ^{-1}\left (\frac {x}{\lambda }\right )}{2 a b}\right )\right \}\right \} \]
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';t:='t'; v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = (c*arccot(x/lambda)+k*arccot(y/beta))*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \left ( {\frac {{\lambda }^{2}+{x}^{2}}{{\lambda }^{2}}} \right ) ^{1/2\,{\frac {\lambda \,c}{a}}} \left ( {\frac {{\beta }^{2}+{y}^{2}}{{\beta }^{2}}} \right ) ^{1/2\,{\frac {\beta \,k}{b}}}{{\rm e}^{1/2\,{\frac {1}{ab} \left ( -2\,cx\arctan \left ( {\frac {x}{\lambda }} \right ) b-2\,\arctan \left ( {\frac {y}{\beta }} \right ) aky+bx\pi \, \left ( c+k \right ) \right ) }}} \]
____________________________________________________________________________________
Added March 9, 2019.
Problem Chapter 4.7.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c \arccot (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcCot[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right ) \exp \left (\frac {c \left (a \log \left (a^2 \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )\right )+2 \beta (b x-a y) \tan ^{-1}(\beta y+\lambda x)+2 x (a \lambda +b \beta ) \cot ^{-1}(\beta y+\lambda x)\right )}{2 a (a \lambda +b \beta )}\right )\right \}\right \} \]
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';t:='t'; v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*arccot(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \left ( {\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{\lambda }^{2}{x}^{2}+1 \right ) ^{{\frac {c}{2\,a\lambda +2\,b\beta }}}{{\rm e}^{1/2\,{\frac { \left ( -2\,a \left ( \beta \,y+\lambda \,x \right ) \arctan \left ( \beta \,y+\lambda \,x \right ) +x\pi \, \left ( a\lambda +b\beta \right ) \right ) c}{ \left ( a\lambda +b\beta \right ) a}}}} \]
____________________________________________________________________________________
Added March 9, 2019.
Problem Chapter 4.7.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = a x \arccot (\lambda x+\beta y) w \]
Mathematica ✓
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*D[w[x, y], y] == a*x*ArcCot[lambda*x + beta*y]*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[ \left \{\left \{w(x,y)\to c_1\left (\frac {a y-b x}{a}\right ) \exp \left (\frac {a \left (-\beta (a y-b x) \log \left (a^2 \left (\frac {\beta ^2 (a y-b x)^2}{a^2}+\frac {2 \beta \lambda x (a y-b x)}{a}+\lambda ^2 x^2+1\right )+2 a b \beta x \left (\frac {\beta (a y-b x)}{a}+\lambda x\right )+b^2 \beta ^2 x^2\right )+a \left (\frac {\beta ^2 (a y-b x)^2}{a^2}-1\right ) \tan ^{-1}\left (\frac {\beta (a y-b x)}{a}+\frac {b \beta x}{a}+\lambda x\right )+a \lambda x+b \beta x\right )}{2 (a \lambda +b \beta )^2}+\frac {1}{2} x^2 \cot ^{-1}\left (\frac {\beta (a y-b x)}{a}+\frac {b \beta x}{a}+\lambda x\right )\right )\right \}\right \} \]
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';t:='t'; v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = a*x*arccot(lambda*x+beta*y)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); sol:=simplify(sol);
\[ w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \left ( {\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{\lambda }^{2}{x}^{2}+1 \right ) ^{-1/2\,{\frac { \left ( ya-bx \right ) a\beta }{ \left ( a\lambda +b\beta \right ) ^{2}}}}{{\rm e}^{1/4\,{\frac {-2\, \left ( \left ( -{\beta }^{2}{y}^{2}+{\lambda }^{2}{x}^{2}+1 \right ) a+2\,bx\beta \, \left ( \beta \,y+\lambda \,x \right ) \right ) a\arctan \left ( \beta \,y+\lambda \,x \right ) + \left ( \pi \,{\lambda }^{2}{x}^{2}+2\,\beta \,y+2\,\lambda \,x \right ) {a}^{2}+2\,\pi \,ab\beta \,\lambda \,{x}^{2}+\pi \,{b}^{2}{\beta }^{2}{x}^{2}}{ \left ( a\lambda +b\beta \right ) ^{2}}}}} \]
____________________________________________________________________________________
Added March 9, 2019.
Problem Chapter 4.7.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccot ^n(\lambda x)w_y = \left ( c \arccot ^m(\mu x) + s \arccot ^k(\beta y) \right ) w \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*ArcCot[lambda*x]^n*D[w[x, y], y] == (c*ArcCot[mu*x]^m + s*ArcCot[beta*y]^k)*w[x, y]; 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';t:='t'; v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde := a*diff(w(x,y),x)+ b*arccot(lambda*x)^n*diff(w(x,y),y) =(c*arccot(mu*x)^m+s*arccot(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -\int \!{\frac {b \left ( \pi /2-\arctan \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) {{\rm e}^{\int ^{x}\!{\frac {1}{a} \left ( c \left ( \pi /2-\arctan \left ( {\it \_b}\,\mu \right ) \right ) ^{m}+s \left ( \pi /2-\arctan \left ( \beta \, \left ( \int \!{\frac {b \left ( \pi /2-\arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{n}}{a}}\,{\rm d}{\it \_b}-\int \!{\frac {b \left ( \pi /2-\arctan \left ( \lambda \,x \right ) \right ) ^{n}}{a}}\,{\rm d}x+y \right ) \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]
____________________________________________________________________________________
Added March 9, 2019.
Problem Chapter 4.7.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccot ^n(\lambda y)w_y = \left ( c \arccot ^m(\mu x) + s \arccot ^k(\beta y) \right ) w \]
Mathematica ✗
ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; ClearAll[g1, g0, h2, h1, h0, f1, f2]; pde = a*D[w[x, y], x] + b*ArcCot[lambda*y]^n*D[w[x, y], y] == (c*ArcCot[mu*x]^m + s*ArcCot[beta*y]^k)*w[x, y]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; sol = Simplify[sol];
\[ \text {\$Aborted} \] 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';t:='t'; v:='v';q:='q';p:='p';l:='l';g1:='g1';g2:='g2';g0:='g0'; h0:='h0';h1:='h1';h2:='h2';f2:='f2';f3:='f3'; pde := a*diff(w(x,y),x)+ b*arccot(lambda*y)^n*diff(w(x,y),y) =(c*arccot(mu*x)^m+s*arccot(beta*y)^k)*w(x,y); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[ w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac {a\int \! \left ( \pi /2-\arctan \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( {\rm arccot} \left ({\it \_b}\,\lambda \right ) \right ) ^{-n}}{b} \left ( c \left ( \pi /2-\arctan \left ( \mu \, \left ( \int \!{\frac { \left ( \pi /2-\arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-n}a}{b}}\,{\rm d}{\it \_b}-{\frac {a\int \! \left ( \pi /2-\arctan \left ( y\lambda \right ) \right ) ^{-n}\,{\rm d}y}{b}}+x \right ) \right ) \right ) ^{m}+s \left ( \pi /2-\arctan \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}}} \]