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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \frac {2 a b c_1\left (y-\frac {b x}{a}\right )+a \beta k \log \left (a^2 \left (\beta ^2+y^2\right )\right )-2 a k y \tan ^{-1}\left (\frac {y}{\beta }\right )+2 b k x \tan ^{-1}\left (\frac {y}{\beta }\right )+2 b k x \cot ^{-1}\left (\frac {y}{\beta }\right )+b c \lambda \log \left (\lambda ^2+x^2\right )+2 b c x \cot ^{-1}\left (\frac {x}{\lambda }\right )}{2 a b}\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\frac {1}{2\,ba} \left ( k\beta \,\ln \left ( {\frac {{\beta }^{2}+{y}^{2}}{{\beta }^{2}}} \right ) a+2\,{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) ba+c\lambda \,\ln \left ( {\frac {{x}^{2}}{{\lambda }^{2}}}+1 \right ) b-2\,cx\arctan \left ( {\frac {x}{\lambda }} \right ) b-2\,\arctan \left ( {\frac {y}{\beta }} \right ) kya+bx\pi \, \left ( c+k \right ) \right ) }\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \frac {c \left (a \log \left (a^2 \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )\right )+2 \beta (b x-a y) \tan ^{-1}(\beta y+\lambda x)+2 x (a \lambda +b \beta ) \cot ^{-1}(\beta y+\lambda x)\right )}{2 a (a \lambda +b \beta )}+c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\frac {1}{ \left ( 2\,a\lambda +2\,\beta \,b \right ) a} \left ( c\ln \left ( {\beta }^{2}{y}^{2}+2\,\beta \,\lambda \,xy+{x}^{2}{\lambda }^{2}+1 \right ) a+ \left ( 2\,{a}^{2}\lambda +2\,ab\beta \right ) {\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) -2\, \left ( a \left ( \beta \,y+x\lambda \right ) \arctan \left ( \beta \,y+x\lambda \right ) -1/2\,x\pi \, \left ( a\lambda +\beta \,b \right ) \right ) c \right ) }\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to a x \left (\frac {\log \left (\beta ^2 y^2+2 \beta \lambda x y+\lambda ^2 x^2+1\right )}{2 \beta y+2 \lambda x}+\cot ^{-1}(\beta y+\lambda x)\right )+c_1\left (\frac {y}{x}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) ={\frac {1}{2\,\beta \,y+2\,x\lambda } \left ( ax\ln \left ( {x}^{2} \left ( {\frac {\beta \,y}{x}}+\lambda \right ) ^{2}+1 \right ) + \left ( \beta \,y+x\lambda \right ) \left ( a\pi \,x-2\,ax\arctan \left ( \beta \,y+x\lambda \right ) +2\,{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) \right ) }\]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \int _1^x\left (\frac {\cot ^{-1}\left (\beta \left (y-\int _1^x\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k}{a}+\cot ^{-1}(\mu K[2])^m\right )dK[2]+c_1\left (y-\int _1^x\frac {b \cot ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) =\int ^{x}\! \left ( {\frac {\pi }{2}}-\arctan \left ( \mu \,{\it \_a} \right ) \right ) ^{m}+{\frac {1}{a} \left ( {\frac {\pi }{2}}-\arctan \left ( {\frac {\beta }{a\lambda } \left ( {\frac {\ln \left ( {{\it \_a}}^{2}{\lambda }^{2}+1 \right ) b}{2}}-{\frac {b\ln \left ( {x}^{2}{\lambda }^{2}+1 \right ) }{2}}+ \left ( -b{\it \_a}\,\arctan \left ( {\it \_a}\,\lambda \right ) +\arctan \left ( x\lambda \right ) bx-{\frac {\pi \, \left ( x-{\it \_a} \right ) b}{2}}+ay \right ) \lambda \right ) } \right ) \right ) ^{k}}{d{\it \_a}}+{\it \_F1} \left ( -{\frac {bx\pi \,\lambda -2\,bx\arctan \left ( x\lambda \right ) \lambda -2\,ya\lambda +b\ln \left ( {x}^{2}{\lambda }^{2}+1 \right ) }{2\,a\lambda }} \right ) \]
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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["Global`*"]; 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]];
\[\left \{\left \{w(x,y)\to \int _1^y\frac {\cot ^{-1}(\lambda K[2])^{-n} \left (\cot ^{-1}(\beta K[2])^k+a \cot ^{-1}\left (\frac {\mu \left (b x-a \int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\cot ^{-1}(\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]+c_1\left (\int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart; 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'));
\[w \left ( x,y \right ) =\int ^{y}\!{\frac {1}{{\rm arccot} \left ({\it \_b}\,\lambda \right )b} \left ( a \left ( {\frac {\pi }{2}}-\arctan \left ( 2\,{\frac {a\mu \,\int \! \left ( \pi -2\,\arctan \left ( {\it \_b}\,\lambda \right ) \right ) ^{-1}\,{\rm d}{\it \_b}}{b}}+\mu \, \left ( -\int \!2\,{\frac {a}{b \left ( \pi -2\,\arctan \left ( \lambda \,y \right ) \right ) }}\,{\rm d}y+x \right ) \right ) \right ) ^{m}+ \left ( {\frac {\pi }{2}}-\arctan \left ( \beta \,{\it \_b} \right ) \right ) ^{k} \right ) }{d{\it \_b}}+{\it \_F1} \left ( -\int \!2\,{\frac {a}{b \left ( \pi -2\,\arctan \left ( \lambda \,y \right ) \right ) }}\,{\rm d}y+x \right ) \]
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