6.3.23 7.3

6.3.23.1 [973] Problem 1

problem number 973

Added Feb. 11, 2019.

Problem Chapter 3.7.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c \arctan \frac {x}{\lambda }+ k \arctan \frac {y}{\beta } \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcTan[x/lambda] + k*ArcTan[y/beta]; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

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

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6.3.23.2 [974] Problem 2

problem number 974

Added Feb. 11, 2019.

Problem Chapter 3.7.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b w_y = c \arctan (\lambda x+\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*ArcTan[lambda*x + beta*y]; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

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

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6.3.23.3 [975] Problem 3

problem number 975

Added Feb. 11, 2019.

Problem Chapter 3.7.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ x w_x + y w_y = a x \arctan (\lambda x+\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*ArcTan[lambda*x + beta*y]; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

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

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6.3.23.4 [976] Problem 4

problem number 976

Added Feb. 11, 2019.

Problem Chapter 3.7.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ a w_x + b \arctan ^n(\lambda x) w_y = c \arctan ^m(\mu x)+s \arctan ^k(\beta y) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = a*D[w[x, y], x] + b*ArcTan[lambda*x]^n*D[w[x, y], y] == a*ArcTan[mu*x]^m + ArcTan[beta*y]^k; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]\right )+\int _1^x\left (\frac {\tan ^{-1}\left (\beta \left (y-\int _1^x\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]+\int _1^{K[2]}\frac {b \tan ^{-1}(\lambda K[1])^n}{a}dK[1]\right )\right ){}^k}{a}+\tan ^{-1}(\mu K[2])^m\right )dK[2]\right \}\right \}\]

Maple ✓

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

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

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6.3.23.5 [977] Problem 5

problem number 977

Added Feb. 11, 2019.

Problem Chapter 3.7.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y], x] + b*ArcTan[lambda*y]^n*D[w[x, y], y] == a*ArcTan[mu*x]^m + ArcTan[beta*y]^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\tan ^{-1}(\lambda K[1])^{-n}dK[1]-\frac {b x}{a}\right )+\int _1^y\frac {\tan ^{-1}(\lambda K[2])^{-n} \left (\tan ^{-1}(\beta K[2])^k+a \tan ^{-1}\left (\frac {\mu \left (b x-a \int _1^y\tan ^{-1}(\lambda K[1])^{-n}dK[1]+a \int _1^{K[2]}\tan ^{-1}(\lambda K[1])^{-n}dK[1]\right )}{b}\right ){}^m\right )}{b}dK[2]\right \}\right \}\]

Maple ✓

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

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

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