6.3.24 7.4

6.3.24.1 [977] Problem 1

problem number 977

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]];
 

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

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'));
 

\[w \left (x , y\right ) = \frac {2 c \,\operatorname {arccot}\left (\frac {x}{\lambda }\right ) x b +2 \,\operatorname {arccot}\left (\frac {y}{\beta }\right ) k y a +\ln \left (\frac {\beta ^{2}+y^{2}}{\beta ^{2}}\right ) a \beta k +c \lambda \ln \left (\frac {x^{2}}{\lambda ^{2}}+1\right ) b +2 f_{1} \left (\frac {a y -b x}{a}\right ) b a}{2 a b}\]

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6.3.24.2 [978] Problem 2

problem number 978

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]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {c \cot ^{-1}\left (\beta y+\lambda K[1]+\frac {b \beta (K[1]-x)}{a}\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]

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'));
 

\[w \left (x , y\right ) = \frac {\ln \left (\beta ^{2} y^{2}+2 \beta \lambda x y +\lambda ^{2} x^{2}+1\right ) c +f_{1} \left (\frac {y a -b x}{a}\right ) \left (2 \lambda a +2 b \beta \right )+2 c \,\operatorname {arccot}\left (\beta y +\lambda x \right ) \left (\beta y +\lambda x \right )}{2 \lambda a +2 b \beta }\]

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6.3.24.3 [979] Problem 3

problem number 979

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'));
 

\[w \left (x , y\right ) = \frac {\ln \left (\left (\beta y +\lambda x \right )^{2}+1\right ) a x +2 \left (\beta y +\lambda x \right ) \left (a x \,\operatorname {arccot}\left (\beta y +\lambda x \right )+f_{1} \left (\frac {y}{x}\right )\right )}{2 \beta y +2 \lambda x}\]

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6.3.24.4 [980] Problem 4

problem number 980

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'));
 

\[w \left (x , y\right ) = \frac {f_{1} \left (-\frac {2 b \,\operatorname {arccot}\left (\lambda x \right ) x \lambda -2 y a \lambda +b \ln \left (\lambda ^{2} x^{2}+1\right )}{2 a \lambda }\right ) a +\int _{}^{x}\left (\operatorname {arccot}\left (\mu \textit {\_a} \right )^{m} a +{\left (\frac {\pi }{2}-\arctan \left (\frac {\left (2 \,\operatorname {arctanh}\left (\frac {\textit {\_a}^{2} \lambda ^{2}}{\textit {\_a}^{2} \lambda ^{2}+2}\right ) b -2 \,\operatorname {arctanh}\left (\frac {\lambda ^{2} x^{2}}{\lambda ^{2} x^{2}+2}\right ) b -i \pi \sqrt {\textit {\_a}^{2} \lambda ^{2}+2}\, \sqrt {\frac {1}{\textit {\_a}^{2} \lambda ^{2}+2}}\, b +i \pi \sqrt {\lambda ^{2} x^{2}+2}\, \sqrt {\frac {1}{\lambda ^{2} x^{2}+2}}\, b +2 \lambda \left (b \,\operatorname {arccot}\left (\lambda \textit {\_a} \right ) \textit {\_a} -b \,\operatorname {arccot}\left (\lambda x \right ) x +y a \right )\right ) \beta }{2 a \lambda }\right )\right )}^{k}\right )d \textit {\_a}}{a}\]

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6.3.24.5 [981] Problem 5

problem number 981

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'));
 

\[w \left (x , y\right ) = \frac {\int _{}^{y}\frac {a \left (\frac {\pi }{2}+\arctan \left (\frac {\mu \left (\int \frac {1}{\operatorname {arccot}\left (\lambda y \right )}d y a -\int \frac {1}{\operatorname {arccot}\left (\lambda \textit {\_b} \right )}d \textit {\_b} a -b x \right )}{b}\right )\right )^{m}+\operatorname {arccot}\left (\beta \textit {\_b} \right )^{k}}{\operatorname {arccot}\left (\lambda \textit {\_b} \right )}d \textit {\_b}}{b}+f_{1} \left (-\frac {a \int \frac {1}{\operatorname {arccot}\left (\lambda y \right )}d y}{b}+x \right )\]

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